Preheat the oven to 350°F (175°C).
Steam the broccoli florets until just tender, about 4-5 minutes. Drain and set aside.
In a large mixing bowl, combine the condensed cream of mushroom soup, mayonnaise, sour cream, chopped onion, garlic powder, black pepper, and salt. Mix well until smooth.
Add the cooked chicken and steamed broccoli to the bowl and gently fold to coat evenly with the sauce mixture.
Stir in 1/2 cup of the shredded cheddar cheese and 1/2 cup of the shredded mozzarella cheese.
Transfer the mixture to a greased 9×13-inch casserole dish and spread evenly.
In a small bowl, combine the crushed crackers or breadcrumbs with the melted butter. Sprinkle this mixture evenly over the casserole.
Top the casserole with the remaining 1/2 cup cheddar cheese and 1/2 cup mozzarella cheese.
Bake uncovered in the preheated oven for 30-35 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.