Preheat the oven to 350°F (175°C). Grease a 9×13-inch casserole dish with butter or cooking spray.
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 an additional 1 minute until fragrant. Remove from heat.
In a large mixing bowl, combine the cooked rice, shredded chicken, steamed broccoli, sautéed onion and garlic, sour cream, condensed cream of chicken soup, milk, salt, black pepper, and paprika. Stir well to combine.
Fold in 3/4 cup of the shredded cheddar cheese and 3/4 cup of the shredded mozzarella cheese into the mixture.
Transfer the mixture to the prepared casserole dish and spread evenly.
Sprinkle the remaining cheddar cheese, 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.