In a large pot of boiling salted water cook the pasta according to the packaging information. Drain and reserve ½ cup of the cooking water.
Meanwhile in a large deep skillet, heat ¼ cup of the oil. Add the cauliflower and Brussels sprouts and season with salt and pepper. Cover and cook over medium high heat, stirring occasionally, until charred and crispy tender, about 5 minutes.
Add ¼ cup of oil to the skillet and add the onion, garlic, anchovies, crushed red peppers (optional), rosemary, thyme. Cook stirring until the onion is slightly softened, about 3 minutes. Cover and cook over low heat until the cauliflower and sprouts are tender, about 3 minutes. Keep warm.
In a small skillet add the remaining 2 tablespoon olive oil and add the bread crumbs. Cook over moderately high heat, stirring until golden and crisp, about 4 minutes, season with salt and transfer to a bowl.
Add the pasta and reserved cooking water to the vegetables and cook over moderate heat, tossing until the water is nearly absorbed. Remove from the heat and stir the ½ cup of grated Parmigiano Reggiano. Serve the pasta in large bowls, passing the bread crumbs and the cheese alongside.