Preheat the oven to 350°F (175°C).
Steam the broccoli florets for 4-5 minutes until tender but still crisp. Drain and set aside.
In a large mixing bowl, combine the condensed cream of mushroom soup, sour cream, mayonnaise, chopped onion, minced garlic, salt, and black pepper. Stir until smooth.
Add the shredded cooked chicken, steamed broccoli, and 1/2 cup of shredded cheddar cheese to the bowl. Mix gently to combine all ingredients evenly.
Transfer the mixture into a 9×13 inch baking dish and spread it out evenly.
Sprinkle the remaining 1/2 cup of shredded cheddar cheese over the top.
In a small bowl, mix the crushed Ritz crackers with the melted butter until the crumbs are evenly coated.
Sprinkle the buttered cracker crumbs evenly over the cheese layer.
Bake uncovered in the preheated oven for 35-40 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.