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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.
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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.
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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).
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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.
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5
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, then transfer pasta to a large bowl. Gradually add 1 1/2 oz. Parmesan, tossing constantly until you have a glossy, emulsified sauce.
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6
Serve pasta topped with more Parmesan and red pepper flakes and a drizzle of oil.
