Heat 2 tablespoons oil in a large nonstick skillet over medium heat. Add the sausage and cook, breaking up the meat, until browned, about 5 minutes. Transfer to a bowl.

Add the remaining 2 tablespoons oil, broccoli, peppers, and garlic to the skillet. Season with salt and pepper. Cook, stirring often, until softened and beginning to brown, about 8 minutes.

Meanwhile, cook the rigatoni in boiling salted water until al dente. Reserve 1 cup cooking water and drain.

Return the pasta and reserved cooking water to the pot over low heat. Add the broccoli mixture, cooked sausage with juices, Parmesan, lemon juice, and red pepper flakes. Stir vigorously until the liquid thickens and coats the pasta, about 2 minutes.

Season with salt and pepper to taste. Serve with additional Parmesan.