Sorafenib Tosylate With or Without Everolimus in Treating Patients With Advanced, Radioactive Iodine Refractory Hurthle Cell Thyroid Cancer

Eligibility Criteria: 1. Central pathology review submission - Patients must have 10 representative hematoxylin and eosin (H\&E) stained thyroid tissue slides OR tumor block available for submission to central pathology review. This review is mandatory prior to registration to confirm eligibility. 2. Measurable disease - Patients must have measurable disease by Response Evaluation Criteria In Solid Tumors (RECIST) criteria, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as ≥ 20 mm with conventional techniques or as ≥ 10 mm with spiral computed tomography (CT) scan. CT must be performed within 28 days of registration. 3. Radioactive iodine (RAI) - refractory disease defined as 1 or more of the following: * Patients who have received greater than 600 mCi of radioactive iodine in their lifetime OR * RAI-avid metastatic lesion which remained stable in size or progressed despite RAI treatment within 9 months of RAI treatment OR * 10% or more increase in serum thyroglobulin (on thyroid-stimulating hormone \[TSH\]-suppression) within 9 months of RAI treatment OR * Index metastatic lesion non-RAI avid on a diagnostic RAI scan OR * Presence of fluorodeoxyglucose (FDG) avid metastatic lesions on positron emission tomography (PET)/CT scan (standardized uptake values \[SUV\]max \> 5 of any single lesion) 4. Progressive disease defined by RECIST criteria ≤ 14 months 5. Patients must have metastatic disease or locally advanced unresectable disease 6. Prior treatment * Patients may have received prior radiation therapy to index lesions ≥ 28 days prior to registration on this protocol if there has been documented progression by RECIST criteria. Prior radiation therapy to the non-index lesions is allowed if ≥ 28 days prior to registration on this protocol. * Prior RAI therapy is allowed if ≥ 90 days prior to registration 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). * Prior chemotherapy is allowed if ≥ 28 days prior to registration on this protocol. * Patient may have received any number of prior lines of therapy. * No prior use of sorafenib or an mammalian target of rapamycin (mTOR) (including phosphoinositide 3-kinase \[PI3k\] or protein kinase B \[AKT\]) inhibitor for the treatment of thyroid cancer. 7. No history of major surgery ≤ 28 days of registration 8. No history of intracranial brain metastasis 9. Cardiovascular disease. No history of any of the following ≤ 6 months of registration: * Myocardial infarction or unstable angina * New York Heart Association grade III or greater congestive heart failure * Cerebrovascular accident * Grade 3 or 4 peripheral ischemia * Grade 3 or 4 thromboembolic event 10. Liver disease: No history of the following: * Child Pugh Class B or C liver disease * "Chronic active" hepatitis defined as: 1. Hepatitis B surface antigen (HBsAg) \> 6 months 2. Serum hepatitis B virus (HBV) deoxyribonucleic acid (DNA) 20,000 IU/ml (105 copies/ml), lower values 2,000-20,000 IU/ml (104-105 copies/ml) are often seen in hepatitis B e antigen (HBeAg)-negative chronic hepatitis B 3. Persistent or intermittent elevation in alanine aminotransferase (ALT)/aspartate aminotransferase (AST) levels 4. Liver biopsy showing chronic hepatitis with moderate or severe necroinflammation 11. No history of gastrointestinal fistula or gastrointestinal perforation \< 90 days of registration. 12. No known history of prolonged QT syndrome 13. No Grade 3 or 4 hypertension (systolic blood pressure \[BP\] \>160 and or diastolic BP \> 100) that cannot be controlled with medication prior to registration. 14. Concomitant medications: * Chronic concomitant treatment with strong inhibitors of cytochrome P450 3A4 (CYP3A4) is not allowed on this study. Patients on strong CYP3A4 inhibitors must discontinue the drug for 14 days prior to registration on the study. * Chronic concomitant treatment with strong CYP3A4 inducers is not allowed. Patients must discontinue the drug 14 days prior to the start of study treatment. * Patients requiring anticoagulation must be on stable dose of medication prior to registration. 15. Not pregnant and not nursing, because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown. Therefore, for women of childbearing potential only, a negative serum pregnancy test done ≤ 7 days prior to registration is required. 16. Age ≥ 18 years 17. Eastern Cooperative Oncology Group (ECOG) Performance Status ≤ 2 18. Required Initial Laboratory Values: * Absolute neutrophil count (ANC) ≥ 1,500/mm\^3 * Platelet count ≥ 100,000/mm\^3 * Creatinine ≤ 1.5 mg/dL OR * Calculated creatinine clearance ≥ 30 mL/min * Total bilirubin ≤ 1.5 x upper limits of normal (ULN) * Serum glutamic oxaloacetic transaminase (SGOT) (AST) ≤ 2.5 x ULN * Fasting serum cholesterol ≤ 300 mg/dL 19. Documentation of disease: Histologic Documentation - Eligible patients must have histopathologically confirmed Hürthle cell thyroid cancer by central review.