Bring a large pot of salted water to a boil.
Add the pasta and cook until al dente.
Add the broccoli to the boiling pasta for the last 3 minutes of cooking.
Reserve 1 cup of pasta water, then drain pasta and broccoli together.
In a large skillet, heat olive oil and butter over medium heat.
Add diced onion and cook until softened.
Add minced garlic and stir for 30 seconds.
Pour in heavy cream and chicken broth.
Simmer gently for 2–3 minutes.
Remove from heat and whisk in Parmesan and mozzarella until smooth.
Add Italian seasoning, garlic powder, salt, and pepper.
Stir in the pasta and broccoli until coated.
Fold in shredded rotisserie chicken.
Add small amounts of reserved pasta water until the sauce reaches your preferred consistency.
Serve warm with extra Parmesan on top if desired.