Preheat the oven to 350°F (175°C).
Bring a pot of water to a boil and blanch the broccoli florets 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 cook for 3-4 minutes until softened.
Add the minced garlic to the skillet and cook for another 1 minute until fragrant. Remove from heat.
In a large mixing bowl, combine the cream of chicken soup, sour cream, milk, salt, black pepper, and paprika. Stir until smooth.
Add the cooked rice, shredded rotisserie chicken, blanched broccoli, sautéed onion and garlic mixture, and half of the cheddar and mozzarella cheeses to the bowl. Mix until well combined.
Lightly grease a 9×13-inch baking dish. Transfer the mixture into the dish and spread evenly.
Sprinkle the remaining cheddar and mozzarella cheeses along with the grated Parmesan cheese evenly over the top.
Bake uncovered for 30-35 minutes, or until the casserole is bubbly and the cheese is melted and golden brown.
Remove from the oven and let it rest for 5 minutes before serving.