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 sauté until translucent, about 3-4 minutes.
Add the minced garlic to the skillet and cook for 1 minute until fragrant. Remove from heat.
In a large mixing bowl, combine the cooked rice, shredded chicken, steamed broccoli, sautéed onions and garlic, cream of mushroom soup, sour cream, milk, salt, black pepper, and paprika. Mix well until fully combined.
Stir in 3/4 cup of shredded cheddar cheese and 3/4 cup of shredded mozzarella cheese into the mixture.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
Sprinkle the remaining 1/4 cup cheddar cheese, 1/4 cup 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.