Cook pasta. In a large pot, bring 4-6 quarts of salted water to a boil and cook the pasta according to package directions. Drain the pasta and return it to the pan off the heat. Stir in 2 tablespoons butter.
Cook sausage. While the pasta is cooking, in a large skillet over medium heat, cook sausage in 2 tablespoons oil, breaking the sausage apart with a wooden spoon or spatula until crumbled and browned, about 8-10 minutes. Add garlic and oregano, stirring garlic and cooking for an additional minute.
Add broccoli and cream. Add broccoli, chicken broth, the remaining 2 tablespoons of oil, salt, and pepper to the sausage mixture. Cook, occasionally stirring, until broccoli is crisp-tender, about 4-5 minutes. Add the cream and warm for an additional minute or two.
Combine with pasta and add cheese. Transfer the broccoli sausage mixture to the large pot with the pasta and stir, adding ½ cup shredded Parmesan. Taste for seasoning and add additional salt and pepper if desired.
Serve. Serve immediately, using the extra ¼ cup Parmesan to garnish each plate with additional Parmesan cheese.