Bring a large pot of salted water to boil. Add the penne pasta and cook according to package directions until al dente. Toss in the broccoli florets for the last three minutes of cooking.
Once cooked, reserve 1 cup of pasta water, then drain the rest.
In a skillet, heat the olive oil and butter over medium-low heat. Sauté the diced onion for 4-5 minutes until softened. Add the minced garlic and cook for an additional 30 seconds until fragrant.
Pour in the heavy cream and low-sodium chicken broth. Simmer for 2-3 minutes until it slightly thickens.
Remove the skillet from heat and stir in the Parmesan and mozzarella cheeses, whisking until the sauce is smooth. Season with Italian seasoning, garlic powder, salt, pepper, and red pepper flakes, if using.
Toss the drained penne and broccoli in the creamy sauce until fully coated. Gently fold in the shredded rotisserie chicken.
If the sauce is too thick, adjust the consistency with the reserved pasta water. For a restaurant-quality finish, mix in a cold knob of butter just before serving.
Notes
Consider topping with freshly grated Parmesan or an additional dash of red pepper flakes before serving.