Preheat the oven to 350°F (175°C).
In a large mixing bowl, combine the cooked rice, shredded chicken, and steamed broccoli florets.
In a separate bowl, whisk together the condensed cream of chicken soup, sour cream, milk, garlic powder, onion powder, salt, and black pepper until smooth.
Pour the soup mixture over the rice, chicken, and broccoli. Stir gently to combine everything evenly.
Add the shredded cheddar cheese and mozzarella cheese to the mixture and fold them in.
Transfer the mixture into a greased 9×13-inch baking dish and spread it out evenly.
In a small bowl, mix the melted butter with the crushed buttery crackers.
Sprinkle the cracker mixture evenly over the top of the casserole.
Sprinkle the grated Parmesan cheese over the cracker topping.
Bake the casserole in the preheated oven for 40 to 45 minutes, or until the top is golden brown and the casserole is bubbly.
Remove from the oven and let it rest for 5 minutes before serving.