Combine ground beef, egg, panko bread crumbs, onion, and 1/4 cup cream of mushroom soup in a large bowl. Divide beef mixture evenly into 6 portions; shape each portion into a 1/2-inch thick patty.
Heat a large non-stick skillet over medium-heat. Cook the burgers until browned, 3 to 4 minutes on each side. Whisk together the remaining cream of mushroom soup and beef broth in a bowl until smooth. Pour mixture over patties, then reduce heat to low. Simmer until the patties are no longer pink in the center, about 20 minutes.
