Testing the Addition of the Immunotherapy Drug Pembrolizumab to the Usual Chemotherapy Treatment (Paclitaxel and Carboplatin) in Stage III-IV or Recurrent Endometrial Cancer

Inclusion Criteria: * Measurable stage III, measurable stage IVA, stage IVB (with or without measurable disease) or recurrent (with or without measurable disease) endometrial cancer. * Pathology report showing results of institutional MMR IHC testing. * Histologic confirmation of the original primary tumor is required (submission of pathology report(s) is required). Patients with the following histologic types are eligible: Endometrioid adenocarcinoma, serous adenocarcinoma, dedifferentiated/undifferentiated carcinoma, clear cell adenocarcinoma, mixed epithelial carcinoma, adenocarcinoma not otherwise specified (N.O.S.). * Submission of tumor specimens for centralized MMR IHC testing is required after Step 1 and before Step 2 registration. * In patients with measurable disease, lesions will be defined and monitored by RECIST version (v) 1.1. Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded). Each lesion must be \>= 10 mm when measured by CT or magnetic resonance imaging (MRI). Lymph nodes must be \>= 15 mm in short axis when measured by CT or MRI. * Patients may have received * NO prior chemotherapy for treatment of endometrial cancer OR * Prior adjuvant chemotherapy (e.g., paclitaxel/carboplatin alone or as a component of concurrent chemotherapy and radiation therapy \[with or without cisplatin\]) provided adjuvant chemotherapy was completed \>= 12 months prior to STEP 2 regi…