This randomized Phase II trial will compare the progression-free survival (PFS) of sorafenib and everolimus versus sorafenib alone in patients with radioactive iodine refractory hurthle cell thyroid cancer. Prior studies have shown that the median PFS is generally around 4.5 months for sorafenib alone in this disease population. It is hoped that the combination of everolimus and sorafenib can increase the median PFS to at least 9 months. In addition to PFS, this trial will also compare the confirmed response rate, overall survival (OS) and adverse event rates between sorafenib and everolimus vs. sorafenib alone. The primary and secondary objectives for the study are listed below.