Preheat the oven to 350°F (175°C).
Bring a pot of water to a boil and blanch the broccoli florets for 3 minutes until bright green and slightly tender. Drain and set aside.
In a large skillet, melt the butter over medium heat. Add the chopped onion and minced garlic and sauté for 3-4 minutes until the onion is translucent and fragrant.
In a large mixing bowl, combine the cooked rice, shredded chicken, blanched broccoli, sautéed onion and garlic, sour cream, condensed cream of mushroom soup, milk, shredded cheddar cheese, salt, black pepper, and paprika. Mix well until all ingredients are evenly incorporated.
Transfer the mixture into a greased 9×13 inch baking dish and spread evenly.
Sprinkle the shredded mozzarella cheese and grated Parmesan cheese evenly over the top of the casserole.
Bake uncovered in the preheated oven for 30 minutes or until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.