Place the cream of chicken soup, whole milk, and American cheese in a large pot over medium-low heat, stirring occasionally. Add the broccoli and cook for about 8 minutes.
Place the cooked rice in the bottom of a 9x13-inch baking dish. Pour the broccoli mixture over the rice and top with shredded Colby cheese.
Place the crushed crackers and melted butter in a bowl and mix well until the crackers are well coated. Sprinkle the buttered cracker crumbs over the casserole.
Bake the casserole in the preheated oven for about 10 to 15 minutes or until the crackers turn golden brown.