Bring a large pot of salted water to boil. Add penne pasta and cook according to package directions. Three minutes before pasta is done, add broccoli florets to the boiling water. Reserve 1 cup pasta water before draining.
While pasta cooks, heat olive oil and butter in a large skillet over medium-low heat. Add diced onion and cook for 5 minutes until softened. Add minced garlic and cook for 30 seconds, stirring constantly.
Pour heavy cream and chicken broth into the skillet. Bring to a gentle simmer for 2-3 minutes with small bubbles around the edges.
Remove skillet completely from heat. Whisk in Parmesan and mozzarella until completely melted and smooth. Season with Italian seasoning, garlic powder, red pepper flakes, salt, and pepper.
Add drained pasta and broccoli to the sauce, tossing with tongs until evenly coated. Fold in shredded rotisserie chicken.
Add reserved pasta water 2 tablespoons at a time if needed to achieve a glossy, creamy consistency. Serve immediately with extra Parmesan.
