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 skillet, melt the butter over medium heat. Add the chopped onion and minced garlic and sauté until softened and fragrant, about 3-4 minutes.
In a large mixing bowl, combine the cooked rice, cooked chicken, steamed broccoli, sautéed onion and garlic, cream of mushroom soup, sour cream, milk, salt, black pepper, and paprika. Stir until well combined.
Stir in 1/2 cup of the shredded cheddar cheese and 1/2 cup of the shredded mozzarella cheese into the mixture.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
Sprinkle the remaining cheddar and mozzarella cheese evenly over the top of the casserole.
Bake uncovered in the preheated oven for 35-40 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.