Bring a large pot of salted water to a boil. Cook the penne pasta according to package directions until al dente. With about 3 minutes left, toss in the fresh or frozen broccoli florets. Reserve 1 cup of pasta water before draining.
In a large skillet, heat olive oil and butter over medium-low heat. Sauté the diced onion for 4-5 minutes until softened. Add the minced garlic and cook until fragrant, about 30 seconds.
Pour in the heavy cream and chicken broth, and simmer gently for 2-3 minutes.
Remove the skillet from heat and stir in the Parmesan and mozzarella until smooth. Season with Italian seasoning, garlic powder, salt, pepper, and add red pepper flakes if desired.
Add the drained pasta and broccoli to the skillet, tossing well to coat. Gently fold in the shredded rotisserie chicken.
Add the reserved pasta water, 2 tablespoons at a time, until you reach your desired creaminess. Stir in a bit of cold butter before serving for extra richness.
Notes
Great served with a garnish of extra Parmesan cheese and fresh herbs. Leftovers can be refrigerated for up to 3 days or frozen for 2 months.