Bring a large pot of salted water to a boil.
While pot of water is heating up, heat a large nonstick skillet over medium-high heat. Add sausage and cook for about 5 to 7 minutes, or until sausage is fully cooked, and remove from heat.
Once water begins to boil, add the pasta and allow to cook for about 3 minutes. Bring the pasta water back to a boil and add broccoli to the pasta. Continue to cook the pasta and broccoli together for another 12-15 minutes, stirring occasionally. Reserve about 1 cup of the pasta water and set aside. Drain pasta and broccoli together in a strainer.
Return the pot to the stove and set heat to high; add 1 tbsp olive oil, when hot, add garlic. Cook until golden, about 1 minute, reduce heat to low and add pasta, broccoli and sausage back to the pot.
Add in 1/2 cup of the reserved pasta water, the milk and grated parmesan cheese. Mix well, breaking up any large pieces of broccoli.
Season with 1 tsp salt & 1/2 tsp pepper (can add more to taste). Can also add in any remaining reserved pasta water if the mixture seems dry.