Cosibelimab for CSCC in Patients With Kidney Transplant or Hematologic Malignancy

Inclusion Criteria: * COHORT A: Participant must have a history of kidney transplant (at least 6 months prior to enrollment). (A history of more than one kidney transplantation is permitted.) * COHORT B: Participant must have a diagnosis of a hematologic malignancy, including chronic myeloproliferative neoplasm (CMN) or an indolent non-Hodgkin's lymphoma (NHL), or multiple myeloma (MM). * Examples of indolent non-Hodgkin's lymphomas (NHL), including but not limited to: * Chronic lymphocytic leukemia (CLL) * Small cell lymphocytic lymphoma (SLL) * Follicular lymphoma (FL) * Lymphoplasmacytic lymphoma * Waldenström macroglobulinemia * Marginal zone lymphoma * Mantle cell lymphoma, indolent variety * Cutaneous T-cell lymphoma (mycosis fungoides and Sézary syndrome) * Examples of chronic myeloproliferative neoplasms (CMN), including but not limited to: * Chronic myelogenous leukemia (CML) * Polycythemia vera (PV) * Primary myelofibrosis * Essential thrombocythemia (ET) * Chronic neutrophilic leukemia * Chronic eosinophilic leukemia * Examples of myeloma, including but not limited to: * Multiple myeloma (MM) * Smoldering myeloma * Participants must have histologically or cytologically confirmed diagnosis of cutaneous squamous cell carcinoma (CSCC). * Note: Mixed histology is acceptable (e.g. basosquamous, squamous cell carcinoma with adnexal differentiation), and other histologic variants (sarcomatoid carcinoma, spindle cell carcinoma, poorly differentiated carcinoma with squamous features). * Participant must have CSCC that is not amenable to surgery or radiation therapy, or recurrent despite prior surgery or radiation therapy, including locally advanced unresectable CSCC for which local therapy (surgery and/or radiation therapy) is not recommended. This includes CSCC occurring in patients who are not surgical candidates due to comorbidities and patients who refuse surgery (due to concerns about morbidity, disfigurement, etc). * Participants must have measurable disease as per PET Response Criteria in Solid Tumors version * (PERCIST 1.0),46 defined as any tumor (of any size) whose SUV lean (SUL) peak is greater than 1.5 times the mean SUL in the liver + 2 times its standard deviation. Mean SUL in the liver is measured in a 3-cm diameter spherical volume of interest (VOI), which is a measurement of the SUV background. * Age ≥18 years. Because no dosing or adverse event data are currently available on the use of cosibelimab in participants \<18 years of age, children are excluded from this study. * ECOG performance status 0-2 (see Appendix A for definitions of this and Karnofsky Performance Status). * Participants must meet the following organ and marrow function as defined below: * Absolute neutrophil count ≥1.0 K/mcL * Platelets ≥30 K/mcL * Total bilirubin ≤ 1.5 institutional upper limit of normal (ULN), except subjects with Gilbert's syndrome, who must have total bilirubin \< 3.0 mg/dL * AST(SGOT) ≤ 2.5 × institutional ULN * ALT(SGPT) ≤ 2.5 × institutional ULN * Adequate renal function, defined as defined as estimated glomerular filtration rate (eGFR) ≥ 30 mL/min * Serum creatinine ≤ 3.0 x institutional ULN or ≤ 3.0 x baseline (grade ≤ 2 per CTCAE) * Participants with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy. * PD-1 and PD-L1 immune checkpoint antibodies are classified as pregnancy class D. For this reason, and because other therapeutic agents used in this trial are potentially teratogenic, women of child-bearing potential (WOCBP) and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation, and 120 days after completion of cosibelimab administration. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 120 days after completion of cosibelimab administration. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Prior treatment within the past 12 months with any of the following classes of drugs: anti-PD-1 inhibitors, anti-PD-L1 inhibitors, or anti-CTLA-4 inhibitors. * Prior treatment within the past 6 months with CAR-T cell therapies, other cellular therapies, or multiagent cytotoxic chemotherapy regimens (e.g. R-CHOP). (Bispecific antibodies and EGFR- targeted therapies are permitted without a washout period.) * Participants who have adverse events due to prior anti-cancer therapy not recovered to Grade 1 or less, with the exception of alopecia, sensory neuropathy, and cytopenias. * Active autoimmune disease that is currently requiring systemic steroid treatment with prednisone \>10mg daily (or equivalent), anti-CD20 antibodies, intravenous immunoglobulin (IVIG) or JAK inhibitors (ruxolitinib, tofacitinib, upadacitinib, etc). * Severe interstitial lung disease (ILD), or a history of pneumonitis that has required oral or IV steroids in the past 5 years. * Participants who are receiving any other investigational agents for cancer within 2 weeks of study enrollment. * Participants who have uncontrolled, symptomatic infections, or infections requiring systemic antibiotics (prophylactic antimicrobials are permitted). * Participants who have uncontrolled or unstable brain metastases (or other CNS metastases) for whom systemic steroids are required. These patients are excluded because steroids may interfere with the effectiveness of immune checkpoint therapy. * Uncontrolled or significant cardiovascular disease. * Current need for dialysis (hemodialysis or peritoneal dialysis). * History of stem cell transplant or bone marrow transplant * Psychiatric illness or social s