Separate the broccoli florets from the stems and then pare the stems with a knife. Leave the floret clusters whole unless they’re especially large; if so, cut them in half.
Bring a large pot of salted water to a boil over high heat. Add the stems and cook for 3 or 4 minutes; add the florets, and cook for another 5 min. until both the stems and the florets are just tender (test with a sharp paring knife). With tongs or a slotted spoon, transfer the pieces to ice water as they’re done. When cool, drain the broccoli well and chop it coarsely. Top up the pot with more hot water and return it to a boil.
Heat the olive oil in a 12-inch skillet over moderate heat. Add the sausage and cook, crumbling it with a wooden spoon, until it loses its pink color. Add the red pepper flakes and garlic and sauté briefly to release the garlic’s fragrance. Add the broccoli and season well with salt. Stir to coat the broccoli with the seasonings. Keep warm over low heat.
Add the pasta to the boiling water and cook it until it’s just shy of al dente (1 to 2 min. less than the suggested cooking time). Reserve 1 cup of the cooking liquid by removing it with a ladle or a measuring cup. Drain the pasta and return it to the warm, empty pot. Add the contents of the skillet to the pasta and cook it over moderate heat, stirring constantly, until the pasta is al dente, adding as much of the reserved water as needed to keep the pasta moist. Remove from the heat, stir in 1/4 cup of the cheese, and portion the pasta into six warm bowls. Top each portion with some of the remaining cheese.