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 minced garlic and sauté until softened, about 3 minutes.
Whisk in the flour and cook for 1 minute, stirring constantly to make a roux.
Gradually whisk in the milk and cook, stirring constantly, until the sauce thickens, about 3-5 minutes.
Remove the saucepan from heat and stir in the mayonnaise, salt, pepper, and paprika until smooth.
In a large mixing bowl, combine the cooked rice, cooked chicken, steamed broccoli, cheddar cheese, mozzarella cheese, and the prepared sauce. Mix well to combine evenly.
Transfer the mixture into the prepared casserole dish and spread evenly.
Sprinkle the grated Parmesan cheese evenly over the top.
Bake uncovered for 30-35 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from the oven and let rest for 5 minutes before serving.