Preheat the oven to 375°F (190°C). Lightly grease a 9x13-inch baking dish.
Bring a pot of water to a boil. Add the broccoli and cook for 2–3 minutes, until slightly tender. Drain well.
In a large bowl, combine the cream of chicken soup, sour cream, chicken broth, garlic powder, onion powder, paprika, black pepper, and salt.
Add the cooked rice, chicken, and broccoli to the creamy mixture.
Stir in 1 cup of the cheddar cheese until everything is evenly combined.
Transfer the mixture to the prepared baking dish and spread evenly.
Sprinkle the remaining cheddar cheese, mozzarella, and Parmesan over the top.
Cover loosely with foil and bake for 20 minutes.
Remove the foil and bake for another 10–15 minutes, until the casserole is hot, bubbling, and the cheese is golden.
Let the casserole rest for 5 minutes before serving.