The purpose of this study is to find out what effects, good and/or bad, the combination of sorafenib and everolimus will have on your thyroid cancer. Treatment guidelines from the National Comprehensive Cancer Network include sorafenib as a treatment option for thyroid cancer. Sorafenib is pill that is approved by the FDA for the treatment of kidney and liver cancers. Sorafenib may work in many different ways. It helps decrease the blood supply to tumors. By doing so, it may limit the tumor's source of oxygen and nutrients and prevent the tumor from growing. Everolimus is an oral medication that is FDA approved for the treatment of kidney cancer. It inhibits a protein kinase called mTOR ("mammalian Target of Rapamycin"). In laboratory studies, the addition of everolimus to sorafenib works better than sorafenib alone. These two drugs are being used together to treat other types of cancer in other clinical studies. In addition, the cancer will be evaluated to help us find factors that can help predict who would benefit most from this combination of drugs.

Inclusion Criteria:

• Patients must have histopathologically confirmed at MSKCC thyroid carcinoma.
• Patient must agree to allow 2 biopsies of any malignant lesion that can be accessed by ultrasound (e.g., cervical lymph node, peripheral nodule) or without the aid of radiology (i.e., skin lesion)
• Patients must have surgically inoperable and/or recurrent/metastatic disease.
• Patients must have a PET scan prior to the protocol start date and have at least one FDG-avid lesion that has not been removed surgically or radiated (unless it has progressed by RECIST criteria after the completion of radiation therapy and is still FDG-avid). FDG-avidity will be defined as any focus of increased FDG uptake greater than normal activity with SUV maximum levels greater than or equal to 3. PET scan can have been done at any time prior to the start of therapy, although it is recommended that it be done within 3 months prior to the start of therapy.
• Patients must have measurable disease by RECIST criteria, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions) as ≥ 20 mm with conventional techniques or as ≥ 10 mm with spiral CT scan. For malignant lymph nodes, the short axis must be > 15 mm when assessed by CT scan; performed ≤ 4 weeks of protocol start date.
• Patients must have progressive disease defined by at least one of the following occurring during or after previous treatment (including RAI treatment) unless newly diagnosed:
• The presence of new or progressive lesions on CT/MRI.
• New lesions on bone scan or PET scan.
• Rising thyroglobulin level (documented by a minimum of three consecutive rises, with an interval of > 1 week between each determination).
• Prior RAI therapy is allowed if > 3 months prior to initiation of therapy on this protocol and evidence of progression (as defined above) has been documented in the interim. A diagnostic study using <10 mCi of RAI is not considered RAI therapy.
• Patients may have received prior external beam radiation therapy to index lesions ≥ 4 weeks prior to initiation of therapy on this protocol if there has been documented progression by RECIST criteria. Prior external beam radiation therapy to the non-index lesions is allowed if ≥ 4 weeks prior to initiation of therapy on this protocol.
• ECOG performance status ≤ 2 (or Karnofsky performance status ≥ 60%).
• Patients must have normal organ and marrow function as defined below:
• Absolute neutrophil count ≥1,500/mcL
• Hemoglobin > 9 gm/dl
• Platelets ≥100,000/mcL
• Total bilirubin ≤ 1.5 X institutional ULN (unless increase is due to indirect bilirubin only such as Gilbert's disease)
• AST(SGOT)/ALT(SGPT) ≤ 2.5 X institutional ULN unless liver metastasis present. or AST(SGOT)/ALT(SGPT) ≤ 5 X institutional ULN if liver metastasis is present.
• Creatinine ≤ 1.5 X institutional ULN OR Creatinine clearance ≥ 60 mL/min/1.73 m2 for patients with creatinine levels above 1.5 X institutional ULN [in this circumstance, either of a measured level based on a 24 hour urine collection, or a calculated level using the Cockcroft and Gault equation: (140 - age in years) X (weight in kg) X (0.85 if female)/72 X serum Cr may be used].
• International normalized ratio (INR) ≤ 1.5 (or in range INR, usually between 2 and 3, if patient is on a stable dose of therapeutic warfarin).

ULN = upper limit of normal

• Fasting serum cholesterol ≤300 mg/dL OR ≤7.75 mmol/L AND fasting triglycerides ≤ 2.5 x ULN. NOTE: In case one or both of these thresholds are exceeded, the patient can only be included after initiation of appropriate lipid lowering medication.
• Ability to understand and the willingness to sign a written informed consent document.
• Age 18 years old or older.

Exclusion Criteria:

• Anaplastic thyroid carcinoma present in any biopsy or fine needle aspirate specimen over the previous year, unless a pathologist at MSKCC disagrees with this diagnosis.
• Previous treatment with a known mTOR inhibitor (e.g., everolimus, temsirolimus) or sorafenib for thyroid cancer.
• Patients currently receiving anticancer therapies or who have received anticancer therapies within 4 weeks of the start of study drug (including chemotherapy, radiation therapy, antibody based therapy, etc.)
• Patients, who have had a major surgery or significant traumatic injury within 4 weeks of start of study drug, patients who have not recovered from the side effects of any major surgery (defined as requiring general anesthesia) or patients that may require major surgery during the course of the study
• Patients receiving chronic, systemic treatment with corticosteroids or another immunosuppressive agent. Topical or inhaled corticosteroids are allowed.
• Patients should not receive immunization with attenuated live vaccines within one week of study entry or during study period
• Patients may not be receiving any other investigational agents.
• Patients with known history of active intraparenchymal brain metastasis within previous 3 months. Previously treated lesions are eligible if they either have been surgically removed and there are no indications of metastatic disease on imaging of the brain or if there has been no progression after treatment for at least 6 months.
• Serious or non-healing wound, ulcer, or bone fracture.
• History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 28 days of treatment.
• Impairment of gastrointestinal function or gastrointestinal disease that may significantly alter the absorption of RAD001 or sorafenib (e.g., ulcerative disease, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome or small bowel resection)
• Patients with clinically significant cardiovascular disease as defined by the following:
• History of CVA within past 6 months
• Myocardial infarction, CABG or unstable angina within past 6 months
• New York Heart Association grade III or greater congestive heart failure or Canadian Cardiovascular Class grade III or greater angina within past 6 months (Appendices A&B)
• Clinically significant peripheral vascular disease within past 6 months
• Pulmonary embolism, DVT, or other thromboembolic event within past 6 months
• Uncontrolled coronary artery disease, angina, congestive heart failure, or ventricular arrhythmia requiring acute medical management within past 6 months
• History of myocardial infarct, cerebrovascular accident, or transient ischemic event within the past 6 months
• liver disease such as cirrhosis, chronic active hepatitis or chronic persistent hepatitis
• active (acute or chronic) or uncontrolled severe infections
• Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection or psychiatric illness/social situations that would limit compliance with study requirements.
• While the use of Angiotensin-Converting Enzyme (ACE) inhibitors is not absolutely excluded, efforts should be made to see if patients on ACE inhibitors can be taken off the medication or switched to another medication.
• Patients with an active, bleeding diathesis
• Female patients who are pregnant or breast feeding, or adults of reproductive potential who are not using effective birth control methods. If barrier contraceptives are being used, these must be continued throughout the trial by both sexes. Hormonal contraceptives are not acceptable as a sole method of contraception. (Women of childbearing potential must have a negative serum pregnancy test within 14 days prior to administration of RAD001)
• Patients with a known hypersensitivity to RAD001 (everolimus) or other rapamycins (sirolimus, temsirolimus) or to its excipients.
• History of noncompliance to medical regimens
• Patients unwilling to or unable to comply with the protocol
• A known history of HIV seropositivity
• The use of agents that inhibit or induce CYP3A metabolism is not strictly prohibited, but should be avoided if possible. Potential CYP3A inducing agents include carbamazepine, phenytoin, barbiturates, rifabutin, rifampicin, and St. John's Wort. Potential CYP3A inhibitors include protease inhibitors, antifungals, macrolide antibiotics, nefazodone, and selective serotonin inhibitors.