Preheat the oven to 350°F (175°C).
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a large mixing bowl, combine the shredded rotisserie chicken, cooked rice, and steamed broccoli.
In a separate bowl, whisk together the sour cream, condensed cream of chicken soup, milk, garlic powder, onion powder, salt, and black pepper until smooth.
Pour the sour cream mixture over the chicken, rice, and broccoli mixture. Stir gently to combine everything evenly.
Fold in the shredded cheddar cheese and mozzarella cheese until well incorporated.
Transfer the mixture to a greased 9×13-inch casserole dish and spread it out evenly.
In a small bowl, mix the melted butter with the crushed crackers and grated Parmesan cheese.
Sprinkle the cracker topping evenly over the casserole.
Bake in the preheated oven for 35-40 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.