Preheat the oven to 350°F (175°C). Lightly grease a 9×13 inch casserole dish.
In a medium saucepan, melt the butter over medium heat. Add the diced onion and minced garlic and sauté until the onion is translucent, about 3-4 minutes.
Whisk in the flour and cook for 1 minute to form a roux.
Gradually whisk in the chicken broth and milk, stirring constantly until the mixture thickens and starts to bubble, about 4-5 minutes.
Remove the sauce from heat and stir in 1 cup of shredded cheddar cheese until melted and smooth. Season with salt, black pepper, and paprika.
In a large mixing bowl, combine the cooked chicken, instant rice, broccoli florets, and cheese sauce. Mix well to evenly coat all ingredients.
Transfer the mixture to the prepared casserole dish and spread evenly.
Sprinkle the remaining 1/2 cup cheddar cheese and 1/2 cup mozzarella cheese evenly over the top.
Cover the casserole dish with aluminum foil and bake in the preheated oven for 20 minutes.
Remove the foil and bake for an additional 10 minutes, or until the cheese on top is melted and lightly golden.
Remove from oven and let the casserole rest for 5 minutes before serving.