Bring a large pot of salted water to boil and cook penne pasta according to package directions until al dente. During the last 3 minutes of cooking time, add broccoli florets directly to the pasta water.

Before draining, reserve 1 cup of starchy pasta water in a measuring cup. Drain pasta and broccoli together and set aside.

Heat olive oil and butter in your largest skillet over medium-low heat. Cook diced onion for 4 to 5 minutes until softened, then add minced garlic for 30 seconds until fragrant.

Pour in heavy cream and chicken broth, bringing mixture to a gentle simmer for 2 to 3 minutes. Look for small bubbles around the edges, not a rolling boil.

Remove skillet from heat completely before adding cheese to prevent graininess. Whisk in Parmesan and mozzarella until completely smooth, then season with Italian seasoning, garlic powder, salt, pepper, and red pepper flakes.

Add drained pasta and broccoli to the creamy sauce, tossing gently with tongs to coat everything evenly. Fold in shredded rotisserie chicken last.

Add reserved pasta water 2 tablespoons at a time until you reach a creamy, glossy consistency. Stir in a cold knob of butter just before serving for restaurant-quality finish.