Preheat the oven to 350 degrees F (175 degrees C). Butter the bottom and sides of a 9×13-inch dish.
Bring a pot of water to a boil. Add chicken and continue to boil until no longer pink, 5 to 10 minutes.
Meanwhile, combine broccoli, cream of chicken soup, Cheddar cheese, 1/2 cup Parmesan cheese, mozzarella cheese, sour cream, and pepper in a large bowl. Mix well.
Drain chicken and add to broccoli mixture. Mix well. Pour into the prepared baking dish and spread evenly.
Bake in the preheated oven for 20 minutes. Add remaining Parmesan cheese on top and continue to bake until cheese is melted, 3 to 5 minutes.
You can use thawed, frozen broccoli instead of fresh, if preferred.
You can sprinkle whatever cheese you want on top after baking for 20 minutes.
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