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, 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 until small bubbles appear around the edges.
Remove skillet from heat and whisk in Parmesan and mozzarella until completely smooth. Season with Italian seasoning, garlic powder, salt, pepper, and red pepper flakes.
Add drained pasta and broccoli to the creamy sauce, tossing gently. Fold in shredded rotisserie chicken last.
Stir in reserved pasta water 2 tablespoons at a time until you reach a creamy consistency. Add a cold knob of butter just before serving for a restaurant-quality finish.
Notes
For a healthier option, use less cheese or opt for lower-fat alternatives. Leftovers taste even better the next day!