This phase II trial tests how well sulforaphane works in preventing melanoma in patients with multiple atypical moles (nevi) and a prior history of melanoma. Patients with a history of melanoma and multiple atypical nevi are at a significantly increased risk of developing additional melanomas than those without atypical nevi. Prevention measures usually include sun protection, monthly skin self-examinations and regular skin examinations from a doctor. Sulforaphane is a naturally-occurring substance found in many cruciferous vegetables, including broccoli, that may prevent melanoma from forming.