Preheat the oven to 350°F (175°C).
Bring a pot of water to a boil and blanch the broccoli florets for 2-3 minutes until bright green and slightly tender. Drain and set aside.
In a medium skillet, melt the butter over medium heat. Add the diced onion and sauté for 4-5 minutes until translucent and softened. Remove from heat.
In a large mixing bowl, combine the shredded rotisserie chicken, cooked rice, blanched broccoli, sautéed onions, sour cream, cream of mushroom soup, milk, garlic powder, onion powder, salt, and black pepper. Stir until well combined.
Fold in 1 1/2 cups of shredded cheddar cheese and all the mozzarella cheese into the mixture.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
Sprinkle the remaining 1/2 cup of shredded cheddar cheese and grated Parmesan cheese evenly over the top.
Bake uncovered for 30-35 minutes until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.