Trim the broccoli and cut it into bite-sized pieces. Bring 4 quarts of cold water to a boil in a large pot. Add the broccoli and salt to taste. Cook for 5 minutes. Scoop out the broccoli with a small sieve. Reserve the cooking water in the pot.
Pour the oil into a skillet large enough to hold all the ingredients. Add the garlic and red pepper and cook over medium heat for about 2 minutes, or until the garlic is lightly golden. Add the broccoli and a pinch of salt. Cook, stirring occasionally, for 10 minutes, or until the broccoli is very soft.
Meanwhile, bring the water back to a boil. Add the rigatoni and cook, stirring frequently, until the pasta is not quite tender. Scoop out about 1 cup of the cooking water and set it aside. Drain the rigatoni and add it to the skillet with the broccoli. Add the reserved cooking water and cook, stirring often, for 5 minutes more.
Sprinkle with the cheese, toss, and serve immediately.