Bring a large pot of generously salted water to a boil and drop in the Orecchiette.
Put a large sauté pan over a medium flame and heat the oil. Add the sausage to the pan and sauté while breaking it up in to small pieces with a wooden spoon. Cook until the sausage has nicely browned, about 7-9 minutes.
Add the garlic and red pepper flakes to the pan and sauté for 2-3 minutes. Lower the flame to low.
Drop the broccoli into the pasta water and cook until the broccoli is tender and the pasta is al dente, about 7-8 minutes.
Scoop the pasta and broccoli out of the water and add them to the pan with the sausage. Toss everything together over medium-high heat for 1-2 minutes. The broccoli should break down and begin to coat the pasta.
Serve in bowls with a drizzle of extra virgin olive oil.