Preheat the oven to 375°F (190°C). Lightly grease a 9×13-inch casserole dish.
In a large skillet, melt the butter over medium heat. Add the diced onion and minced garlic and sauté until softened and fragrant, about 3-4 minutes.
Add the diced chicken breasts to the skillet. Season with salt, black pepper, dried thyme, and paprika. Cook until the chicken is no longer pink on the outside but not fully cooked through, about 5 minutes. Remove from heat.
In a large mixing bowl, combine the uncooked rice, chopped broccoli florets, cream of chicken soup, chicken broth, sour cream, shredded cheddar cheese, shredded mozzarella cheese, and the cooked chicken with onions and garlic. Stir well to combine all ingredients evenly.
Pour the mixture into the prepared casserole dish and spread it out evenly.
Cover the casserole dish tightly with aluminum foil and bake in the preheated oven for 40 minutes.
After 40 minutes, remove the foil and bake uncovered for an additional 10 minutes, or until the rice is tender and the cheese on top is melted and slightly golden.
Remove from the oven and let the casserole rest for 5 minutes before serving.