Set up a grill at your tailgate site and bring on the burgers.
In a medium skillet cook mushrooms, green sweet pepper, and onion in hot butter until tender; cool.
In a large bowl combine egg, bread crumbs, Parmesan cheese, ketchup, celery salt, coriander, and cayenne. Add mushroom mixture and ground beef; mix well. Shape into four 3/4-inch-thick patties. Brush onion slices with olive oil.
Grill patties and onion slices on the rack of an uncovered grill directly over medium coals for 14 to 18 minutes or until an instant-read thermometer inserted in the burgers registers 160 degree F, turning once halfway through grilling. Serve burgers topped with onion slices on buns; add cheese and tomato slices, if desired. Makes 4 servings.