This phase III study evaluates whether cemiplimab can shrink or prevent the return of cancer when combined with the usual approach for skin cancer more effectively than the usual approach alone. This study will enroll adults diagnosed with stage III/IV cutaneous squamous cell carcinoma. If eligible to participate in this study participants will be randomly assigned to 1 of 2 groups. Group 1 will receive the usual surgery used to treat this type of cancer. After surgery, they may get the usual type of radiation, depending on the results from the tumor tissue removed during surgery. Group 2 will receive cemiplimab before surgery every 3 weeks for up to 12 weeks. They may also receive radiation after surgery depending on the tumor tissue results from surgery. If treated with cemiplimab before surgery, then tumor tissue results will also determine whether or not they receive cemiplimab every 6 weeks for up to 24 weeks, after surgery, and radiation if given. Participants will be asked to complete quality of life questionnaires to understand how they are feeling during treatment and the effects of the treatment. Participants will be checked for sides effects 3 to 4 months for 2 years after treatment. After that, every 6 months for 1 year and then annually. The main risks are tiredness, diarrhea or constipation, nausea, rash or itching, change in thyroid function, and risk of progression of the cancer to require earlier surgery or to become inoperable. Alternatives to this treatment is standard of care surgery and radiation. The study benefit is cemiplimab may help in shrinking or stabilizing this type of skin cancer.