Bring 6 quarts water to boil in a large spaghetti pot and add 2 tablespoons salt.
In a 14- to 18-in. sauté pan, heat oil until just smoking and add onion, mint and red pepper. Cook until soft, about 5 to 6 minutes.
Meanwhile, remove leaves and base core from cauliflower and chop into ¼-in. pieces. Break the florets away from the central core and size them similarly. Cut the core into ¼-in. coins and throw all the pieces into the sauté pan with the onions. Lower the heat to medium and cook, stirring regularly for about 12 to 15 minutes, until cauliflower is softened and light brown but not mushy.
Cook the rigatoni according to package instructions until “al dente,” about 8 to 12 minutes, and drain in colander. Toss the hot pasta into pan with cauliflower and add grated cheese, black pepper and parsley. Toss to coat and serve immediately.