Fill a large pot with salted water and bring it to a boil. Add the penne pasta and cook according to the package until nearly tender, adding the broccoli florets in the final 3 minutes. Reserve a cup of pasta water before draining the pasta and broccoli together, then set aside.
Warm the olive oil and butter in a large skillet over medium-low heat. Sauté the diced onion for about 4 to 5 minutes until softened, then stir in the minced garlic and cook for another 30 seconds until it becomes fragrant.
Pour in the heavy cream and chicken broth, and gently simmer the mixture for 2 to 3 minutes, just until small bubbles appear around the edges.
Take the skillet off the heat before whisking in the Parmesan and mozzarella cheeses until the sauce is smooth. Season with Italian seasoning, garlic powder, salt, pepper, and optionally red pepper flakes.
Add the cooked pasta and broccoli into the creamy sauce, tossing carefully with tongs to coat everything evenly. Fold in the shredded rotisserie chicken last.
Gradually stir in the reserved pasta water, 2 tablespoons at a time, until the sauce achieves a creamy, glossy finish. Just before serving, stir in a cold knob of butter to add richness and shine.
