Preheat the oven to 350°F (175°C).
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.
Heat olive oil in a skillet over medium heat. Add the chopped onion and minced garlic and sauté for 3-4 minutes until translucent and fragrant. Remove from heat.
In a large mixing bowl, combine the cooked rice, shredded chicken, blanched broccoli, sautéed onion and garlic, cream of mushroom soup, sour cream, milk, salt, black pepper, and paprika. Mix well until all ingredients are evenly incorporated.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
Sprinkle the shredded cheddar cheese and mozzarella cheese evenly over the top of the casserole.
Sprinkle the grated Parmesan cheese over the cheese layer.
Bake uncovered in the preheated oven for 30 minutes, or until the casserole is bubbly and the cheese is melted and golden brown.
Remove from the oven and let it rest for 5 minutes before serving.