Inclusion Criteria:

          -  Patients must be 18 years old or older.

          -  ECOG performance status < 2 (Appendix 1).

          -  Patients must have histologically confirmed differentiated thyroid cancer, that is
             metastatic or unresectable and for which standard curative or palliative measures do
             not exist or are no longer effective.

          -  Patients may have received multiple treatments of radioactive iodine. Patients may
             have received one or more prior MKI treatments that are not lenvatinib (note: all
             patients will have been required to have had lenvatinib for entry, see below). At
             least 4 weeks must have elapsed since prior non-lenvatinib MKI treatment.

          -  Patients are eligible immediately following progression on lenvatinib. Patients must
             have had documented progression while on the prior treatment with lenvatinib and must
             have had a minimum of stable disease for 4 months.

          -  Measurable disease defined as at least one malignant lesion that can be accurately and
             serially measured in at least one dimension (longest diameter to be recorded) using
             conventional methods (CT, MRI, x-ray, PE) (diameter > 20 mm) or spiral CT (diameter >
             10 mm).

          -  Life expectancy greater than 3 months.

          -  Adequate organ function that has been determined within 14 days prior to enrollment,
             defined as:

          -  Leukocyte count > 3,000/uL; Absolute neutrophil count (ANC) > 1,500/mm3, platelets >
             75,000/mm3, and hemoglobin > 9 g/dl.

          -  Serum creatinine < 1.5 times ULN, or 24-hour creatinine clearance > 75 cc/min. (Note:
             creatinine clearance need not be determined if the baseline serum creatinine is within
             normal limits).

          -  Serum bilirubin < 2.0mg/dL; ALT and AST < 2.5 ULN;

          -  INR < 2.0 or a PT/PTT within normal limits. Exception allowed for patients receiving
             anti-coagulation treatment with an agent such as warfarin or heparin who may be
             allowed to participate. For patients on warfarin, the INR should be measured prior to
             initiation of lenvatinib and monitored at least weekly, or as defined by the local
             standard of care, until INR is stable.

          -  Intellectual, emotional, and physical ability to comply with oral medication.

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

          -  Signed informed consent obtained prior to any screening procedures.

        Exclusion Criteria:

          -  Patients who have had any intervening systemic cancer treatment since the prior
             lenvatinib treatment.

          -  Patients with significant medical disease including: uncontrolled congestive heart
             failure; active symptoms of coronary artery disease, uncontrolled seizure disorder;
             active infection; uncontrolled diabetes mellitus; requirement for chronic high dose
             corticosteroid treatment (Topical or inhaled corticosteroids are allowed); requirement
             for concurrent immunosuppressive drug(s); active autoimmune disease.

          -  Patients with organ allografts.

          -  Patients with uncontrolled BP prior to the start of treatment

          -  Patients with known HIV-infection (HIV testing is not required for participation).

          -  Pregnant or nursing (lactating) women;

          -  Women of child-bearing potential (WOCBP), defined as all women physiologically capable
             of becoming pregnant, must use highly effective methods of contraception during the
             study and 8 weeks after. Highly effective contraception methods include any of the
             following:

          -  Use of oral, injected or implanted hormonal methods of contraception or;

          -  Placement of an intrauterine device (IUD) or intrauterine system (IUS);

          -  Barrier methods of contraception: condom or occlusive cap (diaphragm or cervical/vault
             caps) with spermicidal foam/gel/film/cream/ vaginal suppository;

          -  Total abstinence or;

          -  Male/female sterilization. Women are considered post-menopausal and not of
             child-bearing potential if they have had 12 months of natural (spontaneous) amenorrhea
             with an appropriate clinical profile (e.g. age appropriate, history of vasomotor
             symptoms) or have had surgical bilateral oophorectomy (with or without hysterectomy)
             or tubal ligation at least six weeks prior to randomization. In the case of
             oophorectomy alone, only when the reproductive status of the woman has been confirmed
             by follow up hormone level assessment is she considered not of child-bearing
             potential. Male patients whose sexual partner(s) are WOCBP who are not willing to use
             adequate contraception, during the study and for 8 weeks after the end of treatment

          -  Patients with a history of second cancer (except adequately non-melanoma skin cancer,
             in situ treated cancer of the cervix, uterus, colon cancer or melanoma, any benign
             cancer for which the patient does not require treatment, or any other cancer for which
             the patient has been disease-free for three or more years).

          -  Patients who have received live attenuated vaccines within 1 week of start of
             Everolimus and during the study. Patient should also avoid close contact with others
             who have received live attenuated vaccines. Examples of live attenuated vaccines
             include intranasal influenza, measles, mumps, rubella, oral polio, BCG, yellow fever,
             varicella and TY21a typhoid vaccines;

          -  Patients with uncontrolled diabetes mellitus as defined by HbA1c >8% despite adequate
             therapy. Patients with a known history of impaired fasting glucose or diabetes
             mellitus (DM) may be included, however blood glucose and antidiabetic treatment must
             be monitored closely throughout the trial and adjusted as necessary

          -  Patients with known intolerance or hypersensitivity to everolimus or other rapamycin
             analogs (e.g. sirolimus, temsirolimus);

          -  Patients who have any severe and/or uncontrolled medical conditions such as:

          -  unstable angina pectoris, symptomatic congestive heart failure, myocardial infarction
             ≤6 months prior to start of Everolimus, serious uncontrolled cardiac arrhythmia, or
             any other clinically significant cardiac disease

          -  Symptomatic congestive heart failure of New York heart Association Class III or IV

          -  active (acute or chronic) or uncontrolled severe infection, liver disease such as
             cirrhosis, decompensated liver disease, and active and chronic hepatitis (i.e.
             quantifiable HBV-DNA and/or positive HbsAg, quantifiable HCV-RNA),

          -  known severely impaired lung function (spirometry and DLCO 50% or less of normal and
             O2 saturation 88% or less at rest on room air),

          -  active, bleeding diathesis;

          -  Patients requiring chronic treatment with corticosteroids or other immunosuppressive
             agents. Topical or Topical or inhaled corticosteroids are allowed;

          -  Patients with a known history of HIV seropositivity;

          -  Use of any experimental therapy within 4 weeks prior to baseline evaluations done
             prior to enrollment (with the exception of lenvatinib which may be continued until
             treatment start). Therefore, all experimental treatments other than lenvatinib must be
             discontinued 4 weeks prior to baseline studies or enrollment.

          -  Patients with known impairment of gastrointestinal (GI) function or GI disease that
             may significantly alter the absorption;

          -  Patients with a history of non-compliance to medical regimens or who are considered
             potentially unreliable or will not be able to complete the entire study;

          -  Patients already scheduled to have a lymph node dissection may have the procedure
             performed on-study as long as the lymph nodes are not followed as part of RECIST
             criteria and all study medications are held from three days prior to the procedure to
             one week following or until deemed safe by the treating physician

          -  Patients with carcinomatous meningitis are excluded from the study. Patients with
             treated sites of leptomeningeal disease may be included at the discretion of the
             principal investigator.

          -  Patients who are not expected to tolerate a starting dose of 18mg lenvatinib and 5mg
             everolimus daily will not be considered eligible for the study.

          -  Excludedtherapies and medications, previous or concomitant:

          -  Anticancer chemotherapy or immunotherapy during the study or within 4 weeks prior to
             the first dose of the study drug with the exception of lenvatinib

          -  Radiotherapy for the treatment of a symptomatic (e.g. bone metastasis) as clinically
             indicated is allowed as long as it is not evidence of progressive disease and is not
             required to be measured as a target lesion per RECIST Biological response modifiers,
             such as G-CSF, within 3 week prior to study entry. (G-CSF and other hematopoietic
             growth factors may be used in the management of acute toxicity such as febrile
             neutropenia when clinically indicated or at the discretion of the investigator;
             however they may not be substituted for a required dose reduction).

          -  Patients taking chronic erythropoietin are permitted provided no dose adjustment is
             undertaken within 2 months prior to the study or during the study.

          -  Investigational drug therapy outside of this trial during or within 4 weeks prior the
             screening assessment.

          -  Use of ketoconazole, itraconazole, and ritonavir.

          -  Use of carbamazepine, phenytoin, phenobarbital.

          -  Use of grapefruit juice products.

          -  Use of cyclosporin.

          -  Use of coumadin is discouraged, and patients where possible should be switched to
             lovanox.