Bring a large pot of water to a rolling boil. Salt generously.
Cut broccoli crowns into florets, leaving about 1 inch of stem. Add broccoli to boiling water and cook for about 10 minutes, until tender. Using a skimmer, tongs or a slotted spoon, transfer broccoli to a bowl of cold water (do not drain water from pot). Drain broccoli, then chop fine and set aside.
Bring water in pot back to a boil and add pasta. While pasta is cooking pull out about 1 c of cooking water and set aside.
Meanwhile, heat 3 tablespoons olive oil over medium heat in a wide, heavy skillet and add garlic, anchovies and red pepper flakes. Cook, stirring and mashing anchovies with the back of a wooden spoon, until fragrant, about 1 minute.
Stir in chopped broccoli. Add about 1/2 cup of the pasta water to the broccoli, season to taste with salt and pepper and stir over medium heat for 5 minutes. Broccoli should be very soft.
When pasta is cooked according to package directions, drain and toss at once with broccoli. Add the additional pasta water you set aside if you think it needs more moisture. Serve at once, sprinkled with cheese.