1 large head of broccoli (1 1/4 – 1 1/2 lb.), cut into florets. Stalk peeled and coarsely chopped. (I just bought a package of florets)
2 Tbsp. extra-virgin olive oil, plus more for drizzling
4 garlic cloves, smashed
12 oz. fresh sweet Italian sausage (about 3 links) casings removed
Crushed red pepper flakes
16 oz. orecchiette
3 Tbsp. unsalted butter, cut into pieces
1 1/2 cups Parmesan, finely grated, plus more for serving if desired
Cook broccoli in a large pot of salted boiling water until crisp-tender, about 3 minutes. Using a slotted spoon, transfer broccoli to a colander and let cool (save pot of water for cooking pasta). Chop broccoli into small pieces, set aside.
Heat 2 Tbsp. oil in a large skillet over medium heat. Cook garlic, shaking skillet occasionally, until it starts to turn golden, about 2 minutes. Add sausage and a generous pinch of red pepper flakes and break meat into smaller pieces with a wooden spoon. Cook, stirring occasionally and continuing to break up sausage until it’s browned and cooked, 6-8 minutes.
Bring reserved pot of water to a boil and cook pasta until barely al dente, about 9 minutes (set a timer for 3 minutes less than the package instructions. It will cook more in the skillet.)
Meanwhile, ladle about 1/2 cup pasta cooking liquid from pot into skillet with sausage and add blanched broccoli. Keep mixture at a low simmer, stirring often and mashing with spoon to break up sausage even more, until pasta is finished cooking.
Using a spider or slotted spoon transfer pasta to skillet, then ladle in 1/2 cup pasta water. Cook, stirring, until pasta absorbs most of the liquid and is just al dente, about 4 minutes. Add butter and stir until melted. Gradually add Parmesan, tossing constantly until you have a glossy, emulsified sauce.