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 unsalted butter over medium heat. Add the chopped yellow onion and minced garlic, sautéing until softened and fragrant, about 3-4 minutes.
In a large mixing bowl, combine the cooked white rice, shredded chicken, steamed broccoli, sautéed onion and garlic, condensed cream of mushroom soup, sour cream, milk, salt, black pepper, and paprika. Mix until well combined.
Stir in 1/2 cup of shredded sharp cheddar cheese and 1/2 cup of shredded mozzarella cheese into the mixture.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
Sprinkle the remaining 1/2 cup sharp cheddar cheese and 1/2 cup mozzarella cheese evenly on top.
Bake uncovered in the preheated oven for 30 minutes, or until the casserole is bubbly and the cheese on top is melted and golden.
Remove from the oven and let rest for 5 minutes before serving.