Preheat the oven to 350°F (175°C).
Steam the broccoli florets until tender but still crisp, about 4-5 minutes, then drain and set aside.
In a large mixing bowl, combine the cooked rice, shredded chicken, steamed broccoli, cream of mushroom soup, sour cream, milk, garlic powder, onion powder, salt, and black pepper. Mix until well combined.
Stir in the shredded cheddar cheese and half of the shredded mozzarella cheese into the mixture.
Lightly grease a 9×13-inch baking dish with melted butter.
Transfer the mixture into the prepared baking dish and spread evenly.
Sprinkle the remaining mozzarella cheese and grated Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 30-35 minutes, or until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from the oven and let it rest for 5 minutes before serving.