Step 1Bring a large pot of liberally salted water to a boil. While the water is heating, clean the broccoli: Separate the florets from the stems and set them aside. Remove the tough peel of the stems with a paring knife or vegetable peeler, then cut the stems into half-inch cubes. Remove the casings from the sausage.
Step 2Warm the olive oil in a large skillet over medium-low heat and crumble the sausage meat into the oil, flattening each piece slightly so it will brown better. Cook until the sausage browns and crisps. Add the garlic and red pepper flakes and cook until the garlic turns pale brown, about 5 minutes. Do not let the garlic scorch; if it begins to brown, remove the pan from the heat.
Step 3Add the pasta and the diced broccoli stems to the boiling water and cook 6 to 7 minutes. Add the broccoli florets and cook until the pasta is tender, about 10 minutes total. Scoop out the pasta and broccoli and drain; keep the water at a boil.
Step 4Place the skillet over high heat and add the drained broccoli and pasta to the sausage and garlic. Add one-half cup boiling water from the pasta pot. Cook, stirring frequently, until the water mostly boils away, about 1 1/2 to 2 minutes. Season to taste with salt and a good grinding of black pepper.
Step 5Divide the pasta evenly among 6 warmed pasta plates and shave pecorino Romano over the top with a vegetable peeler.