In a large bowl, combine the eggs, onion, worcestershire sauce, salt and pepper. Crumble beef over mixture and mix well. Shape into 12 thin patties. Broil 4 inches from the heat for 7-8 minutes on each side or until no longer pink.
In a small bowl, combine the cheese, mayonnaise, mustard and relish; mix well. Spoon 2 tablespoons on each burger. Return to the broiler just unil cheese softens. Serve on buns with lettuce, onion and tomato.