Preheat the oven to 350°F (175°C). Grease a 9×13 inch baking dish with butter or non-stick spray.
Bring a medium pot of water to a boil. Add the broccoli florets and cook for 2-3 minutes until bright green and slightly tender. Drain and set aside.
In a medium skillet, melt the butter over medium heat. Add the chopped onion and minced garlic and sauté for 3-4 minutes until softened and fragrant. Remove from heat.
In a large mixing bowl, combine the cooked rice, shredded chicken, blanched broccoli, sautéed onion and garlic, sour cream, cream of chicken soup, milk, salt, black pepper, and paprika. Mix well until fully combined.
Stir in 1 cup of shredded cheddar cheese and all of the mozzarella cheese into the mixture.
Transfer the mixture into the prepared baking dish and spread evenly.
Sprinkle the remaining 1/2 cup shredded cheddar cheese evenly over the top.
Bake uncovered in the preheated oven for 30-35 minutes until the casserole is bubbly and the cheese on top is melted and golden.
Remove from the oven and let the casserole rest for 5 minutes before serving.