1. 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.
2. Heat 2 Tbsp. oil in a large skillet over medium. 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 up meat into smaller pieces with a wooden spoon. Cook, stirring occasionally and continuing to break up sausage, until it is browned and cooked through, 6–8 minutes.
3. 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).
4. 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.
5. Using a spider or slotted spoon, transfer pasta to skillet, then ladle in ½ 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, then transfer pasta to a large bowl. Gradually add 1 1/2 oz. Parmesan, tossing constantly until you have a glossy, emulsified sauce.
6. Serve pasta topped with more Parmesan and red pepper flakes and a drizzle of oil.
