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 metastati…