Preheat the oven to 350°F (175°C). Lightly grease a 9×13 inch casserole dish.
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 chopped onion and cook until soft and translucent, about 3-4 minutes.
Whisk in the flour and cook for 1-2 minutes to form a roux, stirring constantly.
Gradually whisk in the whole milk, ensuring no lumps form. Continue cooking and stirring until the sauce thickens, about 5 minutes.
Remove the sauce from heat and stir in the sour cream, garlic powder, onion powder, salt, and black pepper.
Add 1 cup of shredded cheddar cheese and all of the mozzarella cheese to the sauce, stirring until melted and smooth.
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 to the prepared casserole dish and spread evenly.
Sprinkle the remaining 1/2 cup shredded cheddar cheese and grated Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from oven and let rest for 5 minutes before serving.