Inclusion Criteria:

          -  Age >= 18 years. Children are excluded from this study because neither dosing nor
             safety data are currently available for AZD9291 or BC2059 in patients < 18 years of
             age

          -  Pathology-confirmed metastatic NSCLC

          -  A common activating mutation must be present in the EGFR gene, i.e., exon 19 deletion
             or L858R. The presence of uncommon EGFR mutations, e.g., G719X, S768I, or L861Q are
             also permitted if they co-occur with a common activating mutation. Mutation status
             must be determined using a tumor biopsy by local Clinical Laboratory Improvement Act
             (CLIA)-certified assessment. Mutations identified by blood-based testing can be
             provided, but must be verified by tumor biopsy

          -  The presence of a concurrent T790M mutation, while uncommon in patients who are naïve
             to treatment with EGFR TKIs, is also permitted

          -  Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST)
             version 1.1 criteria

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)

          -  The patient must be able to swallow pills

          -  Life expectancy > 3 months

          -  Leukocytes >= 3,000/mcL

          -  Absolute neutrophil count >= 1,500/mcL

          -  Platelets >= 100,000/mcL

          -  Hemoglobin >= 90 g/L

          -  Total bilirubin =< 1.5 x institutional upper limit normal (ULN) and up to 3 mg/dL for
             patients with Gilbert's

          -  Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase
             [SGOT])/alanine aminotransferase ALT (serum glutamate pyruvate transaminase [SGPT]) =<
             2.5 x institutional ULN and =< 5 x institutional ULN for patients with liver
             metastases

          -  Creatinine within 1.5 x ULN OR glomerular filtration rate (GFR) >= 50 mL/min/1.73 m^2
             (measured or calculated by Cockcroft and Gault equation) -confirmation of creatinine
             clearance is only required for patients with creatinine levels above institutional
             upper limit of normal

          -  If evidence of chronic hepatitis B virus (HBV) infection, HBV viral load must be
             undetectable on suppressive therapy if indicated

          -  If history of hepatitis C virus (HCV) infection, it must be treated and have an
             undetectable viral load

          -  Patients with treated brain metastases are asymptomatic and not requiring ongoing
             treatment

          -  Patients with new or progressive brain metastases (active brain metastases) or
             leptomeningeal disease are eligible if the treating physician determines that
             immediate central nervous system (CNS)-specific treatment is not required and is
             unlikely to be required during the first cycle of therapy. Clinical stability on a
             stable dose of decadron is permitted

          -  Patients with a prior or concurrent malignancy whose natural history or treatment will
             not interfere with the safety or efficacy assessment of the investigational drugs are
             eligible for this trial

          -  Patients with known history or current symptoms of cardiac disease, or history of
             treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac
             function using the New York Heart Association Functional Classification. To be
             eligible for this trial, patients must be class 2B or better

          -  Ability to understand and sign a written informed consent document

        Exclusion Criteria:

          -  Prior treatment with an EGFR TKI in any setting

          -  Patients who have not recovered from adverse events (AEs) due to prior anti-cancer
             therapy (i.e., have residual toxicities > grade 1), with the exception of alopecia

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

          -  Patients who are receiving any other investigational agent or immunotherapy within
             five half-lives of the compound or 3 months, whichever is greater

          -  Patients with an uncontrolled intercurrent illness

          -  Patients with psychiatric illness/social situations that would limit compliance with
             study requirements

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition as BC2059 or AZD9291. Patients with hypersensitivity to any of the
             inactive excipients should also be excluded

          -  Currently receiving (or unable to stop use prior to receiving the first dose of study
             treatment) medications or herbal supplements known to be potent inducers of CYP3A4
             (wash-out periods vary. All patients must try to avoid concomitant use of any
             medications, herbal supplements and/or ingestion of foods with known inducer effects
             on CYP3A4

          -  Pregnant women are excluded from this study because BC2059 has the potential for
             teratogenic or abortifacient effects. There is also an unknown but potential risk for
             AEs in nursing infants secondary to treatment of the mother with AZD9291 or BC2059.
             Breastfeeding patients will be excluded

          -  Patients with a significant history of cardiovascular disease (e.g., myocardial
             infarction [MI], thrombotic or thromboembolic event in the last 6 months)

          -  Any of the following cardiac criteria:

               -  Mean resting corrected QT interval (corrected QT [QTc] using Fredericia's formula
                  [QTcF]) > 470 msec. RR is the time from the interval of 1 QRS complex to the next
                  measured in seconds and is commonly calculated as (60/HR)

               -  Any clinically important abnormalities in rhythm, conduction or morphology of
                  resting ECG (e.g., complete left bundle branch block, third degree heart block,
                  second degree heart block)

               -  Any factors that increase the risk of QTc prolongation or risk of arrhythmic
                  events such as decompensated heart failure, hypokalemia, 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) < lower limit of normal (LLN) as
                  assessed by either multigated acquisition (MUGA) scan or echocardiogram (ECHO)

          -  Patients with active malignancies other than NSCLC or patients with prior curatively
             treated malignancy at high risk of relapse during the study period with the exception
             of localized squamous or basal cell skin cancers, ductal carcinoma in-situ (DCIS), or
             indolent cancer currently on observation (i.e. chronic lymphocytic leukemia [CLL] or
             low-risk prostate cancer)

          -  Any evidence of severe or uncontrolled systemic disease, including uncontrolled
             hypertension and active bleeding diatheses, which in the investigator's opinion makes
             it undesirable for the patient to participate in the trial or which would jeopardize
             compliance with the protocol, or active infection with human immunodeficiency virus
             (HIV). Screening for chronic conditions is not required

          -  Patients who are at risk for impaired absorption of oral medication including, but not
             limited to, refractory nausea and vomiting, chronic gastrointestinal diseases,
             inability to swallow the formulated product or previous significant bowel resection
             that would preclude adequate absorption of AZD9291

          -  Judgment by the investigator that the patient should not participate in the study if
             the patient is unlikely to comply with study procedures, restrictions, or requirements

          -  Involvement in the planning and/or conduct of the study (applies to both investigator
             staff and/or staff at the study site)