Preheat the oven to 375°F (190°C).
Bring a large pot of water to a boil. Add the broccoli florets and cook for 3 minutes until tender-crisp. Drain and set aside.
In a large skillet, heat olive oil over medium heat. Add minced garlic and sauté for 1 minute until fragrant.
Add the softened cream cheese, heavy cream, and mayonnaise to the skillet. Stir continuously until the mixture is smooth and heated through.
Add onion powder, salt, and black pepper to the sauce. Stir to combine.
In a large mixing bowl, combine the shredded chicken, cooked broccoli, and cheese sauce. Mix well.
Fold in 1/2 cup of the shredded cheddar cheese and 1/2 cup of the shredded mozzarella cheese into the mixture.
Transfer the mixture to a greased 9×13 inch casserole dish. Sprinkle the remaining cheddar, mozzarella, and Parmesan cheeses evenly on top.
Bake in the preheated oven for 25 minutes, or until the casserole is bubbly and the cheese on top is golden brown.
Remove from oven and let cool for 5 minutes before serving.