In a medium bowl, combine egg, corned beef, cabbage, bread crumbs, caraway seeds, and salt. Add ground beef; mix well. Shape mixture into four 3/4-inch-thick patties.
For a charcoal grill, place patties on the rack of an uncovered grill directly over medium coals. Grill for 14 to 18 minutes or until done (160 degrees F), turning once halfway through grilling. Add onion slices to the grill during the last 10 to 12 minutes or until tender, turning once halfway through grilling. Toast kaiser rolls on the grill. (For a gas grill, preheat grill. Reduce heat to medium. Place patties, then onion slices and kaiser rolls on grill rack over heat. Cover and grill as above.)
Spread kaiser rolls with horseradish mustard. Serve burgers in rolls with onion slices.