Preheat oven to 350 degrees F (175 degrees C). Butter a 9 inch baking dish.
In a medium bowl, whisk together condensed cream of chicken soup, mayonnaise, egg, onion, salt and pepper.
Place broccoli into a very large mixing bowl. Using a rubber spatula, scrape soup-mayonnaise mixture on top of broccoli, and mix well. Sprinkle on cheese and mix well. Spread mixture into prepared baking dish, and smooth top of casserole. Mix the cracker crumbs and butter together, sprinkle over the top of the casserole.
Bake for 45 minutes to 1 hour in the preheated oven.