Preheat the oven to 350°F (175°C).
Bring a medium pot of water to a boil. Add the broccoli florets and cook for 3-4 minutes until tender-crisp. Drain and set aside.
In a medium skillet, melt the butter over medium heat. Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic to the skillet and cook for 1 minute until fragrant. Remove from heat.
In a large mixing bowl, combine the cream of mushroom soup, sour cream, milk, salt, black pepper, and paprika. Stir until smooth.
Add the cooked rice, shredded rotisserie chicken, cooked broccoli, sautéed onion and garlic mixture, 1 cup of cheddar cheese, mozzarella cheese, and Parmesan cheese to the bowl. Mix everything until well combined.
Lightly grease a 9×13-inch baking dish. Transfer the mixture into the dish and spread evenly.
Sprinkle the remaining 1/2 cup cheddar 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.