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, heat olive oil over medium heat. Add the chopped onion and minced garlic and sauté until the onion is translucent, about 3-4 minutes.
In a large mixing bowl, combine the cooked rice, shredded chicken, steamed broccoli, sautéed onion and garlic, cream of mushroom soup, sour cream, milk, salt, black pepper, and paprika. Stir until well combined.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
Sprinkle the shredded cheddar and mozzarella cheese evenly over the top of the casserole.
Bake uncovered in the preheated oven for 30-35 minutes, or until the cheese is melted and bubbly and the casserole is heated through.
Remove from the oven and let it rest for 5 minutes before serving.