Bring a large pot of generously salted water to boil and add your pasta, cooking according to package directions for al dente pasta. Add the broccoli to the pot when there are 8 minutes left to cook your pasta so the broccoli does not overcook. *See notes* Drain pasta and broccoli once cooked, reserving ¼ cup of the pasta water.
Meanwhile, add the olive oil, garlic and peppers to a large skillet and sauté over medium heat until peppers and garlic are slightly browned, approx. 5 minutes. Add the reserved pasta water, your pasta and broccoli to the pan. Sauté everything together, constantly mixing for 1-2 minutes.