Preheat the oven to 350°F (175°C).
Steam the broccoli florets in a steamer basket over boiling water for 5 minutes until tender-crisp. Drain and set aside.
In a large mixing bowl, combine the shredded cooked chicken, steamed broccoli, cream of mushroom soup, mayonnaise, shredded cheddar cheese, chopped onion, garlic powder, black pepper, and salt. Mix well until all ingredients are evenly incorporated.
Transfer the mixture into a 9×13 inch baking dish and spread it out evenly.
In a small bowl, mix the crushed crackers with the melted butter until the crackers are evenly coated.
Sprinkle the buttered cracker mixture evenly over the top of the casserole.
Bake in the preheated oven for 35 to 40 minutes, or until the casserole is bubbly and the cracker topping is golden brown.
Remove from the oven and let it cool for 5 minutes before serving.