A PD-1 Checkpoint Inhibitor (Cemiplimab) for High-Risk Localized, Locally Recurrent, or Regionally Advanced Skin Cancer

Inclusion Criteria: * Histologically confirmed, cutaneous squamous cell carcinoma * Patients must have disease that is deemed potentially resectable, at the time of the start of study, by the treating investigator. The decision to perform surgery on patients must be based on good clinical judgment. Eligible patients for surgical resection must have disease that, in the judgment of the surgeon, is deemed potentially resectable, resulting in free surgical margins * Patients must have measurable disease * Patients must have disease that is considered either: (1) high-risk localized CSCC, (2) locally recurrent CSCC, or (3) regionally advanced CSCC. The criteria specific to each of these populations is listed below * For patients with high-risk localized CSCC, at least two of the following clinical or pathologic high-risk features must be present to be eligible: * Clinical risk factors * Any tumor size \> 2.0 cm in diameter * Tumors \> 1.0 cm in high risk locations, including "mask areas" (central face, eyelids, eyebrow, nose, lips \[cutaneous\], periorbital, chin, mandible, preauricular and postauricular skin/sulci, genitalia, hands, feet, cheek, forehead, scalp, neck and pretibial) * Any rapidly growing and/or symptomatic tumor * Pathologic risk factors * Poorly differentiated histology * Depth \> 6 mm in thickness * Acantholytic / adenoid, adenosquamous, desmoplastic, or metaplastic / carcinosarcomatous histologic subtypes * Invasion beyond subcutaneous fat * Perineural, lymphatic, or vascular involvement * Patients with locally recurrent CSCC, that failed prior surgery, radiation or systemic therapy, are eligible, as long as they have measurable disease and are deemed potentially resectable by the treating investigator * Patients with regionally advanced CSCC, including in-transit, cutaneous, subcutaneous or lymph node metastases are eligible, as long as they have measurable disease and are deemed potentially resectable by the treating investigator * Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * Absolute neutrophil count \>= 1,000 /mcL * Absolute lymphocyte count \>= 500 / mcL * Hemoglobin \>= 8.0 g/dL * Platelets \>= 75,000/mcl * Total bilirubin =\< 1.5 x institutional upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SPGT\]) =\< 3 x institutional upper limit of normal * Creatinine =\< 1.8 mg/dl * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 90 days following completion of therapy. Should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately. A female of child-bearing potential is any woman (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) who meets the following criteria: * Has not undergone a hysterectomy or bilateral oophorectomy; or * Has not been naturally postmenopausal for at least 12 consecutive months (i.e., has had menses at any time in the preceding 12 consecutive months) * Ability to understand and the willingness to sign a written informed consent and comply with surgical resection at end of study and other study-related procedures Exclusion Criteria: * Metastatic disease that is unresectable. Patients with visceral metastases are not eligible. Regionally advanced disease, including in-transit, cutaneous, subcutaneous, or nodal metastases are allowed, if deemed potentially resectable by the investigator * Prior treatment with cemiplimab or any other agent that blocks the PD-1 or PD-L1 pathway * Prior treatment with other immune modulating agents within fewer than 4 weeks, prior to the first dose of cemiplimab. Examples of immune modulating agents include blockers of CTLA-4, 4-1BB, OX-40, therapeutic vaccines, or cytokine therapies * Patients must not be receiving other concomitant biologic therapy, hormonal therapy, chemotherapy, other anti-cancer therapy or any other investigational agents while on this protocol * Radiation therapy, non-cytotoxic agents or investigational agents in the 4 weeks prior to registration * Immunosuppressive systemic corticosteroids equivalent to prednisone 10 mg or greater in the 14 days prior to the first dose of cemiplimab * Any major surgery within 14 days prior to the first dose of cemiplimab. Patients must have recovered from any major complications before registration * Active autoimmune disease requiring systemic treatment in the past 2 years (i.e. use of disease modifying agents or immunosuppressive drugs). Replacement therapy (e.g. thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc) is not considered a form of systemic treatment * History of other prior malignancy in the last five years, with the exception of: adequately treated non-melanoma skin cancers (including multiple primary skin cancers), adequately treated in situ cancer, and other local tumors considered cured by local treatment (including melanoma) * History of allergic reactions attributed to compounds of similar chemical or biologic composition to cemiplimab or any other PD-1 or PD-L1 inhibitor * Uncontrolled, intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness / social situations that would limit compliance with study requirements * Positive pregnancy test, active pregnancy or nursing / breast-feeding, due to the potential for congenital abnormalities and the potential of this regimen to harm nursing infants * History solid organ or bone marrow transplantation