Inclusion Criteria:  Documented informed consent of the participant and/or legally authorized representative  Assent, when appropriate, will be obtained per institutional guideline  Agreement to allow the use of archival tissue from diagnostic tumor biopsies  If unavailable, exceptions may be granted with study principal investigator (PI) approval Age: >= 18 years Eastern Cooperative Oncology Group (ECOG) =< 2  Histologically confirmed mycosis fungoides (MF) or Sezary syndrome (SS). Safety lead-in: >= stage IIB OR >= stage IB-IIA folliculotropic/transformed MF. Phase 2: >= stage IB  Stage of disease according to Tumor-Node-Metastasis-Blood (TNMB) classification Pathology report must be diagnostic or be consistent with MF/SS criteria SS is defined as meeting T4 plus B2 criteria; where the biopsy of erythrodermic skin may only reveal suggestive but not diagnostic histopathologic features, the diagnosis may be based on either node biopsy or fulfillment of B2 criteria For MF where the histological diagnosis by light microscopic examination is not confirmed, diagnostic criteria that been recommended by the International Society of Cutaneous Lymphomas (ISCL) should be used. Measurable disease per Modified Severity Weighted Assessment Tool (mSWAT) and/or Sezary count Baseline skin biopsy taken within 6 months available for central review submission  Without bone marrow involvement: Absolute neutrophil count (ANC) >= 1,500/mm^3 (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated)  NOTE: Growth factor is not permitted within 14 days of ANC assessment unless cytopenia is secondary to disease involvement  With bone marrow involvement: ANC >= 1,000/mm^3 (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated)  NOTE: Growth factor is not permitted within 14 days of ANC assessment unless cytopenia is secondary to disease involvement  Without bone marrow involvement: Platelets >= 100,000/mm^3 (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated)  NOTE: Platelet transfusions are not permitted within 14 days of platelet assessment unless cytopenia is secondary to disease involvement  With bone marrow involvement: Platelets >= 75,000/mm3 (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated)  NOTE: Platelet transfusions are not permitted within 14 days of platelet assessment unless cytopenia is secondary to disease involvement Total bilirubin =< 1.5 x upper limit of normal (ULN) (unless has Gilbert's disease)(performed within 7 days prior to day 1 of protocol therapy unless otherwise stated) Aspartate aminotransferase (AST) =< 2.5 x ULN (unless has Gilbert's disease)(performed within 7 days prior to day 1 of protocol therapy unless otherwise stated) Alanine aminotransferase (ALT) =< 2.5 x ULN (unless has Gilbert's disease)(performed within 7 days prior to day 1 of protocol therapy unless otherwise stated) Creatinine clearance of >= 60 mL/min per 24 hour urine test or the Cockcroft-Gault formula (unless has Gilbert's disease) (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated) If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin (PT) =< 1.5 x ULN (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated) If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated) If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) =< 1.5 x ULN (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated) If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated)  Hepatitis C virus (HCV)*, active hepatitis B virus (HBV) (surface antigen negative), and syphilis (rapid plasma reagin [RPR])  If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed  Meets other institutional and federal requirements for infectious disease titer requirements  Note Infectious disease testing to be performed within 28 days prior to day 1 of protocol therapy Subjects with MF and a history of staphylococcus colonization are eligible provided they continue to receive stable doses of prophylactic antibiotics Women of childbearing potential (WOCBP): negative urine or serum pregnancy test If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required  Agreement by females and males of childbearing potential* to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 3 months after the last dose of protocol therapy  Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for > 1 year (women only)  Exclusion Criteria:  Prior mogamulizumab Any systemic therapy, including monoclonal antibody within 28 days or 5 half-lives (whichever is shorter) of initiating protocol therapy Chemotherapy, radiation therapy, biological therapy, immunotherapy within 21 days prior to day 1 of protocol therapy Any skin-directed therapy within 14 days prior to initiating protocol therapy Any radiation therapy within 21 days prior to initiating protocol therapy  Immunosuppressive medication within 14 days prior to the first dose of study treatment. The following are exceptions to this criterion:  Intranasal, inhaled, topical or local steroid injections (e.g., intra-articular injection) and are on stable dose for at least 28 days Systemic corticosteroids at physiologic doses of < 10 mg/day of prednisone or equivalent Live, attenuated vaccine within 30 days prior to the first dose protocol therapy  Disease free of prior malignancies for >= 5 years with the exception of:  Currently treated squamous cell and basal cell carcinoma of the skin, or Carcinoma in situ of the cervix, or Surgically removed melanoma in situ of the skin (stage 0) with histological confirmed free margins of excision, or Prostate cancer (T1a or T1b using the TNM [tumor, nodes, metastasis] clinical staging system) that has/have been surgically cured, or Any other malignancy that has/have been curatively treated with surgery and/or localized radiation Active infection requiring antibiotics Known hepatitis B or hepatitis C infection Other active malignancy Females only: Pregnant or breastfeeding Prior stem cell transplantation Acute infection requiring systemic treatment Conditions requiring chronic steroid or immunosuppressive treatment that likely need additional steroid or immunosuppressive treatments in addition to the protocol therapy Renal failure requiring hemodialysis or peritoneal dialysis  Unstable cardiac disease as defined by one of the following:  Cardiac events such as myocardial infarction (MI) within the past 6 months NYHA (New York Heart Association) heart failure class III-IV Uncontrolled atrial fibrillation or hypertension Major surgery (as defined by the investigator) within the 28 days prior to the first dose of study treatment  Active or prior documented autoimmune or inflammatory disorders requiring therapy within the past 3 years prior to the start of treatment. The following are exceptions to this criterion:  Vitiligo or alopecia Hypothyroidism (e.g., following Hashimoto syndrome) stable on hormone replacement; or Psoriasis not requiring systemic treatment History of primary immunodeficiency Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures. Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)