Inclusion Criteria:

          -  Patients with stage IV or recurrent/metastatic histologically or cytologically
             confirmed non-squamous NSCLC

          -  NSCLC must harbor an EGFR activating mutation (Exon 21 L858R, Exon 19 deletion)

          -  Progressive disease on osimertinib (AZD9291) given first line

          -  For the dose expansion portion ONLY, patient must: 1) have progression of disease with
             first line osimertinib administered for advanced or metastatic disease as the last
             previous systemic treatment, 2) be treatment naïve for other 3rd generation EGFR-TKI
             (CO-1686) and mTOR inhibitors

          -  Patients must have measurable disease by Response Evaluation Criteria in Solid Tumors
             (RECIST) 1.1, defined as at least one lesion that can be accurately measured in at
             least one dimension >= 10 mm (>= 1 cm) by computed tomography (CT) imaging or magnetic
             resonance imaging (MRI) within 28 days prior to start of protocol therapy; the CT from
             a combined positron emission tomography (PET)/CT may be used if it is of diagnostic
             quality; laboratory parameters are not acceptable as the only evidence of disease

          -  For the dose expansion, no other systemic therapies for advanced/metastatic disease is
             permissible after first line osimertinib

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 1 (Karnofsky >= 70%)

          -  Patients with a prior history of brain metastases are eligible provided:

               -  The brain metastases have been treated; patients with small brain metastases for
                  which radiation or surgery is not indicated, may be eligible on discussion with
                  the study chair. Brain metastases that were managed with first line osimertinib
                  is permissible, provided that the brain metastases are stable on a baseline MRI
                  and for whom radiation or surgical resection is not indicated

               -  The patient is asymptomatic from the brain metastases

               -  Corticosteroids and anti-epileptic drugs prescribed for the management of brain
                  metastases have been discontinued at least 7 days prior to registration

               -  The brain metastases are stable on pre-registration imaging

          -  Absolute neutrophil count (ANC) >= 1.5 x 10^9/L

          -  Hemoglobin >= 90 g/L (or >= 9 g/dL)

          -  Platelets >= 100 x 10^9/L

          -  Calculated creatinine clearance of > 50 mL/min using Cockcroft Gault equation or
             24-hour urine sampling

          -  Total bilirubin =< 1.5 institutional upper limit of normal

          -  Aspartate transaminase (AST) (serum glutamic-oxaloacetic transaminase [SGOT]) and
             alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) =< 2.5 x
             institutional upper limit of normal

          -  Fasting plasma glucose =< 130 mg/dL (or 7.2 mmol/L)

          -  Glycosylated hemoglobin (Hb A1c) =< 7.0%

          -  Fasting triglycerides =< 300 mg/dL (3.42 mmol/L)

          -  Cholesterol =< 300 mg/dL (7.75 mmol/L)

          -  Fridericia's correction formula (QTcF) =< 470 msec

          -  Patients must have had sufficient time between a prior therapy and resolution of
             toxicities from the prior therapies prior to registration as follows:

               -  Prior EGFR inhibitor: A minimum of 7 days must have elapsed from the last dose of
                  the prior EGFR inhibitor and resolution of any drug-related toxicity to =< grade
                  1 except for alopecia

               -  Chemotherapy: A minimum of 21 days must have elapsed from the last dose and
                  resolution of toxicity to =< grade 1 excluding =< grade 2 peripheral neuropathy
                  or alopecia

               -  Surgery: A minimum of 21 days must have elapsed following major surgery and
                  resolution of toxicity to < grade 1 and complete wound healing must have occurred

               -  Radiation: A minimum of 14 days must have elapsed following radiation therapy and
                  resolution of all radiation induced toxicity excluding alopecia

               -  Treatment with an investigational drug: A minimum of 3 months or five half-lives
                  of the compound, whichever is greater, must have elapsed from last treatment date

          -  Ability to understand and the willingness to sign a written informed consent document

          -  Available for follow-up of their disease after treatment until progressive disease is
             documented and resolution of related adverse events until < grade 2

        Exclusion Criteria:

          -  Any serious intercurrent or psychiatric illness that could, in the investigator's
             opinion, potentially interfere with the completion of treatment according to this
             protocol including but not limited to:

               -  Uncontrolled diabetes mellitus (fasting plasma glucose > 130 mg/dL or 7.2 mmol/L)

               -  Past medical history of interstitial lung disease, drug-induced interstitial lung
                  disease, radiation pneumonitis which required steroid treatment, or any evidence
                  of clinically active interstitial lung disease

               -  Active infections

               -  Gastrointestinal disease limiting the ability to swallow oral medications or
                  absorb oral medications including refractory nausea and vomiting, chronic
                  gastrointestinal diseases, inflammatory bowel disease; malabsorption syndromes

               -  Patients with enteric stomata or significant bowel resection

               -  Prior history of corneal ulceration

               -  Patients with any evidence of severe or uncontrolled systemic liver disease
                  including those with known hepatitis B and hepatitis C (excluding treated
                  hepatitis C that has been cured)

               -  Active bleeding diatheses

          -  Significant active cardiovascular or pulmonary disease including:

               -  Uncontrolled hypertension (i.e., systolic blood pressure > 180 mmHg, diastolic
                  blood pressure > 95 mmHg); use of anti-hypertensive agents to control
                  hypertension before cycle 1 day 1 is allowed

               -  Pulmonary hypertension

               -  Uncontrolled asthma or oxygen (O2) saturation < 90% by arterial blood gas
                  analysis or pulse oximetry on room air

               -  Significant valvular disease; severe regurgitation or stenosis by imaging
                  independent of symptom control with medical intervention, or history of valve
                  replacement

               -  Any clinically important abnormalities in rhythm, conduction or morphology of
                  resting electrocardiogram (ECG) e.g. medically significant (symptomatic)
                  bradycardia, complete left bundle branch block, third degree heart block and
                  second-degree heart block or history of arrhythmia requiring an implantable
                  cardiac defibrillator; or within the last 6 months before administration of the
                  first dose of drug:

                    -  Requirement for inotropic support (excluding digoxin) or serious
                       (uncontrolled) cardiac arrhythmia (including atrial flutter/fibrillation,
                       ventricular fibrillation or ventricular tachycardia)

                    -  Placement of a pacemaker for control of rhythm

                    -  Ischemic myocardial event, including angina requiring therapy and artery
                       revascularization procedures

                    -  Ischemic cerebrovascular event, including transient ischemic attack and
                       artery revascularization procedures

                    -  New York Heart Association (NYHA) class III or IV heart failure

                    -  Symptomatic pulmonary embolism or asymptomatic pulmonary emboli within the
                       last 3 months

               -  Baseline prolongation of the rate-corrected QT interval (QTc) (e.g., repeated
                  demonstration of QTcF >= 470 msec (mean value) obtained from 3 ECGs, using the
                  screening clinic ECG machine derived QTc value, or history of congenital long QT
                  syndrome, or torsades de pointes); any factors that increase the risk of QTc
                  prolongation or risk of arrhythmic events such as heart failure, hypokalaemia,
                  congenital long QT syndrome, family history of long QT syndrome or unexplained
                  sudden death under 40 years of age in first degree relatives or any concomitant
                  medication known to prolong the QT interval

               -  Left ventricular ejection fraction (LVEF) by either multigated acquisition (MUGA)
                  or echocardiography (ECHO) less than lower limit of normal

          -  Patients with active malignancies other than NSCLC or prior curatively treated
             malignancy at high risk of relapse during the study period with the exception of
             localized squamous or basal cell skin cancers

          -  Concurrent anti-cancer therapy

          -  History of hypersensitivity attributed to compounds of similar chemical or biologic
             composition to MLN0128 (TAK-228) or osimertinib (AZD9291)

          -  Pregnant women or women who are breast feeding are not eligible for the study; the
             effects of MLN0128 (TAK-228) and osimertinib (AZD9291) on the developing human fetus
             are unknown

          -  Women of non-child bearing potential must be:

               -  Women more than 50 years must be post-menopausal for at least 12 months following
                  the end of all exogenous hormonal treatments OR

               -  Women under 50 years must be postmenopausal for at least 12 months following the
                  end of exogenous hormonal treatments and with LH and FSH levels in the
                  post-menopausal range for the institution OR

               -  Documentation of irreversible surgical sterilization by hysterectomy, bilateral
                  oophorectomy or bilateral salpingectomy but not tubal ligation

          -  Women of child bearing potential must have a negative serum or urine pregnancy test
             within 7 days of registration and must agree to:

               -  Practice 1 highly effective method of contraception and 1 additional effective
                  (barrier) method, at the same time, from the time of signing the informed consent
                  through 120 days (or longer, as mandated by local labeling [e.g., United Surgical
                  Partners International (USPI), Summary of Product Characteristics (SmPC), etc.;])
                  after the last dose of study drug, OR

               -  Agree to practice true abstinence, when this is in line with the preferred and
                  usual lifestyle of the patient; NOTE: Periodic abstinence (e.g., calendar,
                  ovulation, symptothermal, postovulation methods), withdrawal, spermicides only,
                  and lactational amenorrhea are not acceptable methods of contraception; female
                  and male condoms should not be used together

          -  Male patients, even if surgically sterilized (i.e., status postvasectomy) must agree
             to:

               -  Practice highly effective barrier contraception during the entire study treatment
                  period and through 120 days after the last dose of study drug, OR

               -  Practice true abstinence, when this is in line with the preferred and usual
                  lifestyle of the patient; NOTE: Periodic abstinence (e.g., calendar, ovulation,
                  symptothermal, postovulation methods for the female partner), withdrawal,
                  spermicides only, and lactational amenorrhea are not acceptable methods of
                  contraception; female and male condoms should not be used together

               -  Not to donate sperm during the course of this study or within 120 days after
                  receiving their last dose of study drug

          -  Inability to discontinue drugs that are strong cytochrome P450 3A4 (CYP3A4) or P450
             3A5 (CYP3A5), cytochrome P450 2C19 (CYP2C19), and cytochrome P450 2C9 (CYP 2C9)
             inhibitors and/or inducers; and substrates of CYP 3A4/5 or CYP1A2 that are sensitive
             or have a narrow therapeutic window at least three weeks prior to study registration

          -  Patients receiving systemic corticosteroids (either IV or oral steroids, excluding
             inhalers or low-dose hormone replacement therapy) within 1 week before administration
             of the first dose of study drug

          -  Consumption of grapefruit or grapefruit juice is not permitted during the study;
             patients should not consume food or beverages containing the fruit or juice of
             grapefruits or Seville oranges within 7 days before the first dose of study drug and
             throughout the study

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy are ineligible because of the potential for pharmacokinetic interactions with
             MLN0128 (TAK-228) and osimertinib (AZD9291)

          -  No concomitant use of proton pump inhibitors (PPI) is allowed; any prior PPI must be
             discontinued at least one week before receiving MLN0128 (TAK-228)