Preheat the oven to 350°F (175°C). Lightly grease a 9×13 inch casserole dish.
Bring a pot of water to a boil and blanch the broccoli florets for 3 minutes until bright green and slightly tender. Drain and set aside.
In a medium saucepan, melt the butter over medium heat. Add the chopped onion and minced garlic and cook until softened, about 3 minutes.
Whisk in the flour and cook for 1 minute to form a roux.
Gradually whisk in the milk and continue to cook, stirring constantly, until the sauce thickens, about 3-5 minutes.
Remove the saucepan from heat and stir in the sour cream, salt, black pepper, and paprika until smooth.
In a large mixing bowl, combine the cooked rice, shredded chicken, blanched broccoli, half of the cheddar cheese, half of the mozzarella cheese, and the creamy sauce. Mix well to combine.
Transfer the mixture to the prepared casserole dish and spread evenly.
Sprinkle the remaining cheddar and mozzarella cheese evenly over the top.
Bake in the preheated oven for 25-30 minutes, until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from oven and let it rest for 5 minutes before serving.