Testing the Addition of an Immunotherapy Drug, Cemiplimab (REGN2810), Plus Surgery to the Usual Surgery Alone for Treating Advanced Skin Cancer

Inclusion Criteria:

* Pathologically (histologically or cytologically) proven diagnosis of invasive cutaneous squamous cell carcinoma (CSCC) or regional lymph node or in-transit metastasis of CSCC

  * The following CSCC subtypes are eligible according to World Health Organization (WHO) classification if the predominant histology is confirmed CSCC.

    * Spindle cell squamous cell carcinoma (SCC)
    * Squamous cell carcinoma with sarcomatoid differentiation
    * Acantholytic SCC
    * Clear cell SCC
    * Lymphoepithelial carcinoma

      * Note: Keratoacanthoma SCC and Verrucous SCC subtypes are not eligible.
  * For patients with regional metastasis without a primary tumor at screening: a clinical history of CSCC that drains to the involved regional lymph nodes or in-transit metastases in question is required

    * For example, a parotid mass shown to be SCC by cytologic analysis of a fine needle aspirate in a patient with a clinical history of CSCC on the ipsilateral scalp would be eligible
  * For patients with regional metastases without a primary tumor and an ambiguous clinical history: tumor genomic sequencing suggesting a primary tumor of cutaneous origin would be acceptable evidence to establish eligibility
  * NOTE: Tumor genomic sequencing is not required to determine eligibility, but may be part of the routine evaluation of patients with cancers of unknown primary at some institutions. For example, a parotid mass shown to be SCC by cytologic analysis of fine needle aspirate without a primary tumor and an ambiguous clinical history, but with a tumor genomic sequencing assay demonstrating a high tumor mutation burden (≥ 10 mutations/Mb) and/or a high fraction of ultraviolet (UV) related mutations (\> 50% of mutations \[cytosine (C)/thymine (T)\]C \> T or CC \> TT) and/or the presence of "signature 7" mutations would be eligible (Chang 2021)
* Previously untreated or recurrent CSCC
* Clinical American Joint Committee on Cancer (AJCC) 8th Edition (eyelid, head and neck sites) or Union for International Cancer Control (UICC) (non-head and neck sites) stage III or IV
* Primary tumor site must be in the head and neck cutaneous region, other non-head and neck cutaneous regions, or eyelid cutaneous region
* No mucosal squamous cell carcinoma (vermillion lip, nasal, oral, sinonasal, conjunctival, anogenital)
* Tumor must be resectable with curative intent. Note: Tumor with bony skull base invasion and/or skull base foramen involvement (T4b) is not eligible. (Patients with T4b eyelid tumors using UICC Staging, and not involving the brain, are eligible.)
* At least 1 lesion that is measurable by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1
* No definitive clinical or radiologic evidence of distant metastatic disease (M1), visceral and/or distant nodal disease
* Age ≥ 18
* Eastern Cooperative Oncology Group (ECOG) performance status of 0-2
* Not pregnant and not nursing

  * Negative urine or serum pregnancy test (in persons of childbearing potential) within 14 days prior to registration. Childbearing potential is defined as any person who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal
* Absolute neutrophil count (ANC) ≥ 1,000 cells/mm\^3
* Platelets ≥ 75,000 cells/mm\^3
* Hemoglobin ≥ 8.0 g/dl (Note: The use of transfusion or other intervention to achieve hemoglobin \[Hgb\] ≥ 8.0 g/dl is acceptable)
* Creatinine clearance (CrCL) \> 30mL/min by the Cockcroft-Gault formula
* Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) (NOTE: For patients with Gilbert's syndrome, total bilirubin ≤ 3 x ULN. Gilbert's syndrome must be documented appropriately as past medical history.)
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) ≤ 3 x institutional ULN
* No prior systemic therapy for the study cancer (including patients currently receiving immunotherapy for a separate malignancy)
* No prior radiotherapy to the region of the study cancer that would result in cumulative doses of radiation to organs at risk for radiation injury that exceed protocol limitations
* No history of myocardial infarction/unstable angina within the last 6 months
* New York Heart Association functional classification IIb or better (New York Heart Association \[NYHA\] functional classification III/IV are not eligible) (Note: Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association functional classification)
* No active infection requiring systemic antibiotics, antiviral, or antifungal treatments
* No history of allogeneic stem cell transplantation, or autologous stem cell transplantation
* No history of a solid organ transplant (other than corneal transplant)
* No active, known, or suspected autoimmune disease

  * Active or known disease is defined as:

    * Requiring higher than physiologic steroid levels (\> 10mg prednisone/day or equivalent) or
    * Requiring disease-modifying agents or
    * Ongoing or recent (within 5 years prior to registration) evidence of significant autoimmune disease that required treatment with systemic immunosuppressive treatments, which may suggest risk for immune-related adverse events (irAEs)
  * NOTES:

    * Patients meeting the following criteria are not considered immunosuppressed and are eligible to enroll:

      * Patients who require a brief course of steroids (eg, prophylaxis for imaging assessments due to hypersensitivity to contrast agents) are not excluded
      * Patients with type I diabetes mellitus, and endocrinopathies (including hypothyroidism due to autoimmune thyroiditis) only requiring hormone replacement, or skin disorders (such as vitiligo, psoriasis, or alopecia) not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are permitted to enroll
      * Physiologic replacement doses ≤ 10 mg prednisone/day or equivalent allowed, as long as they are not being administered for immunosuppressive intent. Inhaled or topical steroids are permitted
    * Patients with the following immunosuppressed conditions are eligible to enroll:

      * Patients with HIV infection on effective anti-retroviral therapy with undetectable viral load within 6 months prior to registration are eligible
      * Patients with chronic lymphocytic leukemia (CLL) with no history of anti-CLL therapy within 6 months prior to registration are eligible
* No history of interstitial lung disease (eg, idiopathic pulmonary fibrosis, organizing pneumonia)
* No active, noninfectious pneumonitis requiring immune-suppressive therapy
* No active tuberculosis
* No live vaccines within 28 days prior to registration
* No history of allergic reaction to the study agent, compounds of similar chemical or biologic composition to the study agent (or any of its excipients)