Bring a large pot of salted water to a boil, then cook the pasta. Meanwhile, complete the remaining steps below. (When the pasta is al dente, remove it from the water and reserve 1 cup of the pasta water.)
Heat a large sauté pan (12 inches) or cast iron skillet over medium-high heat. Add the olive oil and the Italian sausage, and cook until the sausage is browned and completely cooked through. Remove the sausage with a slotted spoon, leaving any oil behind in the pan.
Add the garlic and sauté until soft and fragrant, about 30 seconds. Add the chicken stock, making sure to scrape any fond off the bottom of the pan, and bring to a boil.
Add the broccoli pieces, cover, and turn the heat down to medium. Cook until the broccoli is soft and tender, about 5 minutes.
Remove the lid and add the butter and Parmesan cheese. Stir until melted into the sauce. Then add the cooked sausage and pasta. If the sauce is too thick, thin it with a bit of the reserved pasta water. Season to taste with salt and pepper.
