Preheat oven to 350°F and grease a 9x13-inch baking dish.
In a large bowl, combine cream of chicken soup, sour cream, mayonnaise, garlic powder, and black pepper until smooth.
Stir in the cooked chicken, blanched broccoli, and 1 1/2 cups of cheddar cheese until everything is well coated.
Pour mixture into the prepared baking dish and spread evenly.
Sprinkle remaining cheddar cheese and Parmesan cheese over the top.
Bake for 25 minutes, then remove from oven and top with French fried onions.
Return to oven and bake an additional 5-7 minutes until onions are golden and casserole is bubbly.
Let stand for 5 minutes before serving.