Cemiplimab Before and After Surgery for the Treatment of High Risk Cutaneous Squamous Cell Cancer

Inclusion Criteria: * Patient must have a known diagnosis of high risk cSCC defined by the following criteria: * Nodal disease with extracapsular extension (ECE) and at least one node \>= 20 mm on the surgical pathology report * In-transit metastases (ITM) defined as skin or subcutaneous metastases that are \> 2 cm from the primary lesion but are not beyond the regional nodal basin * T4 lesion for head and neck CSCC * Perineural invasion (PNI), defined as clinical and/or radiologic involvement of named nerves * Recurrent CSCC, defined as CSCC that arises within the area of the previously resected tumor, or at least one of the following additional features: * \>= N2b disease associated with the recurrent lesion * Nominal \>= T3 (recurrent lesion \>= 4 cm in diameter or minor bone erosion or deep invasion \> 6 mm measured from the granular layer of normal adjacent epithelium) * Poorly differentiated histology and \>= 20 mm diameter of recurrent lesion. The recurrent lesion must be documented to be within the area of the previously resected CSCC by radial measurement of the greatest radius of the final defect, measured from the estimated center of the original surgical wound * Cancer confirmed to be surgically resectable, with surgery evaluation with planned prior to resection * No prior systemic immunotherapy, no prior anti-PD1 therapy * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 * Hemoglobin \>= 9.0 g/dl (within 28 days of cycle 1 day 1) * Absolute neutrophil count (ANC) \>= 1,500/mcL (within 28 days of cycle 1 day 1) * Platelets \>= 100,000/mcL (within 28 days of cycle 1 day 1) * Total bilirubin =\<1.5 institutional upper limit of normal (ULN) (within 28 days of cycle 1 day 1) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 2.5 institutional upper limit of normal (ULN) (within 28 days of cycle 1 day 1) * Albumin \>= 3.0 g/dL (within 28 days of cycle 1 day 1) * Serum creatinine =\< 1.5 x ULN (or calculated creatinine clearance of \>= 50 mL/min using Cockcroft-Gault formula) (within 28 days of cycle 1 day 1) * International normalized ratio (INR) =\< 1.5 (within 28 days of cycle 1 day 1) * Anticoagulation is allowed only with low molecular weight heparin (LMWH). Patient receiving LMW heparin on stable therapeutic dose for more than 2 weeks or with factor Xa level \< 1.1U/mL are allowed on the trial * Clinically significant toxic effect(s) of the most recent prior anti-cancer therapy must be grade 1 or resolved (except alopecia and sensory neuropathy); patients with grade 2 adrenal insufficiency related to prior anti-cancer therapy (defined as requiring medical intervention, such as concomitant steroids) or grade 2 hypothyroidism (defined as requiring hormone replacement therapy) may be enrolled provided that clinical symptoms are adequately controlled and the daily dose is 10 mg or less of prednisone or equivalent. If the patient received major surgery or radiation therapy of \> 30 Gy, they must have recovered from the toxicity and/or complications from the intervention * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * The effects of cemiplimab on the developing human fetus are unknown. For this reason female of child-bearing potential (FCBP) must have a negative serum or urine pregnancy test prior to starting therapy * FCBP and men must agree to use adequate contraception (at least one highly effective method and one additional method of birth control at the same time or complete abstinence) prior to study entry, for the duration of study participation and for at least 6 months following study drug discontinuation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. A female of childbearing potential (FCBP) is a sexually mature woman who: has not undergone a hysterectomy or bilateral oophorectomy or has not been naturally postmenopausal for at least 12 consecutive months (if age \>= 55 years); if the female subject is \< 55 years and she has been naturally postmenopausal for \>= 1 year her reproductive status has to be verified by additional lab tests (\< 20 estradiol OR estradiol \< 40 with follicle stimulating hormone \[FSH\] \> 40 in women not on estrogen replacement therapy) * Patients must agree not to donate blood, sperm/ova while taking protocol therapy and for at least 6 months after stopping treatment * Willingness and ability of the subject to comply with scheduled visits, drug administration plan, protocol-specified laboratory tests, other study procedures, and study restrictions * Evidence of a personally signed informed consent indicating that the subject is aware of the neoplastic nature of the disease and has been informed of the procedures to be followed, the experimental nature of the therapy, alternatives, potential risks and discomforts, potential benefits, and other pertinent aspects of study participation Exclusion Criteria: * Determined not to be a surgical candidate due to medical co-morbidities or extent of disease * Treatment with chronic immunosuppressants (e.g., cyclosporine following transplantation) * Prior organ allograft or allogeneic bone marrow transplantation * Subjects with active or history of immune mediated pneumonitis, colitis, hepatitis, nephritis, or skin reactions as these patients may be at increased risk for developing immune therapy-induced exacerbation or recurrence of their immune mediated disease, potentially delaying surgery * Subjects with a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids and adrenal replacement doses \> 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease * Women who are pregnant or lactating * Uncontrolled intercurrent illness including, but not limited to, human immunodeficiency virus (HIV)-positive subjects receiving combination antiretroviral therapy, ongoing or active infection, symptomatic congestive heart failure (New York Heart Association \[NYHA\] class III or IV), unstable angina pectoris, ventricular arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Other medications, or severe acute/chronic medical or psychiatric condition, or laboratory abnormality that may increase the risk associated with study participation or study drug administration, or may interfere with the interpretation of study results, and in the judgment of the investigator would make the subject inappropriate for entry into this study * Clinical evidence of bleeding diathesis or coagulopathy * Subjects with a history of severe allergic reactions * Patients with prior malignancies, are eligible if they have been disease free for \> 3 years * Patients with prior low-risk non-melanoma skin cancers and in situ carcinomas are eligible provided there was complete removal * Use of other investigational drugs (drugs not marked for any indication) within 28 days or at least 5 half-lives (whichever is longer) before study drug administration * History of severe hypersensitivity reactions to other monoclonal antibodies * Non-oncology vaccines within 28 days prior to starting treatment * Prisoners and subjects who are compulsory detained