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; break it up with a wooden spoon and cook until nicely browned, about 5 to 7 minutes, then remove from heat and set aside.
When the water boils, add the orecchiette and return to a boil; when it comes back to a boil, add the broccoli florets to the same pot and cook both according to the pasta package for al dente.
Just before draining, reserve about 1 cup of the pasta cooking water, then drain the pasta and broccoli in a colander.
Return the empty pot to high heat, add 1 tablespoon of the olive oil, and when hot add the smashed chopped garlic; cook until fragrant and lightly golden, about 1 minute.
Reduce heat to low and add the drained pasta and broccoli back to the pot along with the browned sausage; mix well.
Add the remaining 1 tablespoon olive oil, grated cheese, and salt and pepper to taste; toss and gently press any large broccoli pieces to break them up.
Stir in 1/2 cup of the reserved pasta water and mix until the sauce coats the pasta, adding more reserved water as needed to reach desired consistency.
Serve immediately with extra grated cheese on the side if desired.