Bring a large pot of salted water to a rolling boil.
While the water heats, warm a large nonstick skillet over medium-high heat and add the chicken sausage (casing removed). Break it up with a wooden spoon and cook until browned, about 5–7 minutes, then remove from heat and set aside.
When the water is boiling, add the orecchiette and return to a boil; when it has returned to a boil, add the broccoli florets. Cook both according to the pasta package for al dente.
When the pasta is nearly done, reserve about 1 cup of the pasta cooking water, then drain the pasta and broccoli in a colander.
Return the empty pot to the stove over high heat. Add 1 tablespoon of the olive oil and swirl; when hot, add the garlic and cook until golden, about 1 minute, then reduce heat to low.
Add the drained pasta and broccoli to the pot with the cooked sausage. Mix well to combine, add the remaining 1 tablespoon olive oil and the grated cheese, and season with kosher salt and fresh cracked pepper to taste. Break up any large broccoli pieces as you stir.
Add about 1/2 cup of the reserved pasta water and toss; add more reserved water as needed to create a light sauce that coats the pasta.
Serve the pasta in bowls with additional grated cheese on the side, if desired.
