Oral Tamoxifen vs. TamGel vs. Control in Women With Atypical Hyperplasia, Lobular Carcinoma In Situ, or Increased Breast Cancer Risk

Inclusion Criteria: * Willing to return to enrolling institution for follow-up * Willing to complete required testing * Ability to complete questionnaire by themselves or with assistance * Female (sex that was assigned at birth) * Ipsilateral intact breast with histology confirmation of atypical ductal or lobular hyperplasia, or lobular carcinoma in situ (LCIS), within the last 12 months, whether surgically excised or not.; OR neither AH nor LCIS but increased breast cancer risk defined as either: * Gail model (Breast Cancer Risk Assessment Tool) 5 year breast cancer risk of \>= 3%, or * International Breast Intervention Study model 10 year breast cancer risk of \>= 5%. * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%) * The effects of topical afimoxifene (4-OHT) gel on the developing human fetus at the recommended therapeutic dose are unknown. However, oral tamoxifen is Pregnancy Category D-positive evidence of human fetal risk. For this reason, and because triphenylethylene antiestrogens, including tamoxifen, are known to be teratogenic, women of childbearing potential and their male partners must agree to use at least one effective form of birth control (abstinence is not an allowed method) prior to study entry and for the duration of study participation, and for 2 months following the last dose of study medications (participant can resume oral birth control pills for effective birth control measures after post-treatment biopsy is done). Effective birth control methods during treatment are: copper and Mirena intrauterine device (IUD), diaphragm/cervical cap/shield, spermicide, contraceptive sponge, condoms. Tubal Ligation is an acceptable method of birth control. Women of childbearing potential must have a negative pregnancy test within five days before starting study medications. Should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her study physician immediately. * Willingness to avoid exposing breast skin to natural or artificial sunlight (i.e. tanning beds) for the duration of the study. * Participants must have acceptable organ and marrow function as judged by treating physician's evaluation of baseline laboratory data. * Negative pregnancy (serum or urine) test if of childbearing potential and/or follicle stimulating hormone (FSH) to verify menopausal status. Exclusion Criteria: * Clinically suspicious mass/lesions * Breast cancer in the past 5 years. * Patients with any history of venous thromboembolic disease, regardless of timeframe (history of varicose veins and superficial phlebitis is allowed). * Current pregnancy or lactation. * History of other prior breast cancer-specific therapy within the previous 2 years (chemotherapy, anti-HER2 agents, endocrine agents, everolimus, CDK4-6 inhibitors). * Cytotoxic chemotherapy for any indication in last 2 years. * Prior use of selective estrogen receptor modulator (SERMS) or AIs including tamoxifen, raloxifene, anastrozole, letrozole, or exemestane for prevention or therapy within the past 5 years unless: * Use was less than 6 months duration in the past 5 years and not used in the 1 year prior to enrollment, or * Use was no greater than 2 months duration in the past 1 year and not used in the 6 months prior to enrollment. * Exogenous sex steroid, including oral contraceptive pill use within 1 month prior to pretreatment breast biopsy. Use of vaginally administered estrogens and hormone coated IUD such as Mirena is permitted. * History of any prior ipsilateral breast radiotherapy. Previous unilateral radiation of the contralateral side is allowed. * Skin lesions on the breast that disrupt the stratum corneum (e.g., eczema, ulceration). * History of endometrial neoplasia * Current smoker. Cessation for at least 6 weeks * Current users of potent inhibitors of tamoxifen metabolism. The potent inhibitors of tamoxifen metabolism are: bupropion, cinacalcet, fluoxetine, paroxetine, quinidine. * Participants may not be receiving any other investigational agents within 90 days of enrollment or during this study. * History of allergic reactions to tamoxifen. * Uncontrolled intercurrent illness that in the judgement of the treating physician would make them unsuitable for study participation * Current use of anticoagulation medications. * Patients who are breastfeeding. * Hemoglobin \< 10 g/dL (within 30 days of randomization). * Leukocytes \< 3,000/microliter (within 30 days of randomization). * Platelets \< 100,000/microliter (within 30 days of randomization). * Total bilirubin \> 1.5 x institutional upper limit of normal (ULN) (within 30 days of randomization). * Aspartate aminotransferase (AST) serum glutamic oxaloacetic transaminase (SGOT) \> 1.5 x ULN (within 30 days of randomization). * Alanine aminotransferase (ALT) serum glutamate pyruvate transaminase (SGPT) \> 1.5 x ULN (within 30 days of randomization). * Alkaline phosphatase, S \> 1.5 x ULN (within 30 days of randomization). * Albumin, S \> 1.5 x ULN (within 30 days of randomization). * Protein, total, S \> 1.5 x ULN (within 30 days of randomization). * Creatinine \> 1.5 x ULN (within 30 days of randomization). * Antithrombin III \<80% of normal. * Fibrinogen \>1000 mg/dL. * Patients who are taking any medications, herbal products, or over the counter (OTC) products that are moderate or strong CYP2D6 inhibitors or CYP3A inducers. Patients should also refrain from starting any drug or product with these properties during the study. This is to avoid any potential interactions with tamoxifen or 4-OHT. * Clinically significant arrhythmia requiring ongoing medication for control / treatment, especially those with high risk of QT prolonging effects. * Identification of a clinically suspicious mass on examination.