Preheat the oven to 375°F (190°C). Lightly grease a 9×13-inch casserole dish with cooking spray.
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a medium saucepan, melt the butter over medium heat. Add the diced onion and minced garlic and cook until softened, about 3-4 minutes.
Whisk in the flour and cook for 1-2 minutes to form a roux, stirring constantly to avoid lumps.
Gradually whisk in the whole milk, continuing to stir until the mixture thickens, about 5 minutes.
Remove the sauce from heat and stir in the sour cream, salt, black pepper, paprika, dried thyme, onion powder, and garlic powder until smooth.
Add 1 cup of shredded cheddar cheese and all the mozzarella cheese to the sauce and stir until melted and fully combined.
In a large mixing bowl, combine the cooked rice, shredded chicken, steamed broccoli, and cheese sauce. Mix well to evenly distribute all ingredients.
Transfer the mixture into the prepared casserole dish and spread it out evenly.
Sprinkle the remaining 1/2 cup shredded cheddar cheese evenly over the top of the casserole.
Bake in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from the oven and let it cool for 5 minutes before serving.