Lenvatinib and Pembrolizumab in Differentiated Thyroid Cancers (DTC)

Inclusion Criteria: * Locally recurrent and unresectable and/or distant metastatic differentiated thyroid cancer (DTC), histologically or cytologically confirmed; the diagnosis of DTC includes the following subtypes: papillary thyroid cancer (PTC) (including but not limited to variants such as follicular variant, tall cell, columnar cell, Hurthle cell variant of papillary carcinoma, and poorly differentiated), follicular thyroid cancer (FTC), including insular variant, Hurthle cell carcinoma and poorly differentiated thyroid cancer * Measurable disease meeting the following criteria: * At least 1 lesion of \>= 1.0 cm in the longest diameter for a non-lymph node or \>= 1.5 cm in the short-axis diameter for a lymph node which is serially measurable according to RECIST 1.1 using computerized tomography/magnetic resonance imaging (CT/MRI); if there is only one target lesion and it is a non-lymph node, it should have a longest diameter of \>= 1.5 cm * Lesions that have had external beam radiotherapy (EBRT) or loco-regional therapies such as radiofrequency (RF) ablation must show evidence of progressive disease based on RECIST 1.1 to be deemed a target lesion * For cohort 1 only: evidence of disease progression =\< 14 months prior to registration according to RECIST 1.1, as confirmed by the site study principal investigator (PI) * For cohort 2 only: progressive disease (PD) on lenvatinib per RECIST 1.1 =\< 60 days prior to registration, as confirmed by the site study PI; pati…