Bring a large pot of water to a boil. Meanwhile, remove sausage from casings and break into small pieces. Cut broccoli into florets no larger than 1 inch. Slice garlic thinly.

Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add sausage and cook, breaking it up with a wooden spoon, until deeply browned and crispy in spots, about 8 minutes. Transfer to a plate, leaving the rendered fat in the pan.

Add broccoli to the skillet with a pinch of salt. Cook undisturbed until charred in spots, about 4 minutes, then stir and continue cooking until nearly tender, 3 minutes more. Add garlic and red pepper flakes; cook 1 minute until fragrant.

Salt the boiling water aggressively and add orecchiette. Cook 2 minutes less than package directions. Before draining, scoop out 1 cup of pasta water and set aside. Drain pasta.

Add pasta to the skillet with broccoli. Pour in 3/4 cup pasta water and the remaining olive oil. Cook over medium heat, tossing constantly, until the water reduces to a glossy sauce that coats the pasta, about 2 minutes. Return sausage to the pan, add parmesan, and toss vigorously. Remove from heat.

Divide among bowls. Squeeze fresh lemon over each portion and shower with more parmesan. Serve immediately.