BN-Brachyury, Entinostat, Adotrastuzumab Emtansine and M7824 in Advanced Stage Breast Cancer (BrEAsT)

Background: Breast cancer is the second most common cause of United States (U.S.) cancer deaths in women. Immunotherapy drugs use a person's immune system to fight cancer. Researchers want to see if a new combination of immunotherapy drugs can help treat breast cancer that has gone to places in the body outside of the breast (metastasized). Objective: To learn if a new combination of immunotherapy drugs can shrink tumors in people with metastatic breast cancer. Eligibility: Adults 18 and older who have been diagnosed with metastatic breast cancer, such as Triple Negative Breast Cancer (TNBC) or estrogen receptors (ER)-/progesterone receptors (PR)-/human epidermal growth factor receptor 2 (HER2)+ Breast Cancer (HER2+BC) Design: Participants will be screened with: medical history physical exam disease confirmation (or tumor biopsy) tumor scans (computed tomography, magnetic resonance imaging, and/or bone scan) blood and urine tests electrocardiogram (measures the hearts electrical activity) echocardiogram (creates images of the heart). Participants will be assigned to 1 of 3 groups. The drugs they get will be based on the group they are in. Drugs are given in cycles. Each cycle = 3 weeks. Participants will be seen in clinic every 3 weeks, prior to the start of a new cycle. At each visit, participants will have an clinical exam, have blood drawn and will be asked about any side effects. They will repeat the screening tests during the study. New scans, like a computed tomography (CT) scan, will be done every 6 weeks to see if the treatment is working. All participants will get Bavarian Nordic (BN)-Brachyury. It is 2 different vaccines - a prime and a boost. First the priming vaccines, called MVA-BN-Brachyury help to jump start the immune system. Next the boosting vaccines, called fowlpox virus (FPV)-Brachyury help to keep the immune system going. They are injected under the skin during different cycles. All participants will get M7824 (also known as Bintrafusp alfa), which is an immunotherapy drug. Some participants will get a commonly used drug is HER2+ breast cancer called adotrastuzumab emtansine (also known as T-DM1DM1 or kadcyla). For both, a needle is inserted into a vein to give the drugs slowly. Some participants will take Entinostat weekly by mouth. It is in tablet form. Participants will keep a pill diary. Participants will continue on their assigned treatment until their cancer grows, they develop side effects or want to stop treatment. About 28 days after treatment ends, participants will have a follow-up visit or a telephone call. Then they will be contacted every 3 months for 1 year, then every 6 months for 1 year. They may have more tumor scans or continue treatment.