In a large bowl, combine the broccoli, eggs, mayonnaise, soup, onion, and 2 cups of the grated cheese. Add salt and pepper and put into the prepared casserole dish. Bake for 30 minutes. Remove the casserole from the oven and sprinkle the surface with the cracker crumbs. Top with the remaining cheese. Return to the oven and bake until the crumbs brown slightly and the cheese melts, about 10 minutes.