Heat a grill or grill pan. In a bowl, mix together beef, mayonnaise, shallots, salt, pepper and paprika. Roll into 4 hot dog - shaped logs. Oil the grill pan lightly. Grill burger dogs until charred and thoroughly cooked on all sides. For cheese dog version, place slice of cheddar cheese over burger dog during the last minute of cooking, just so it melts over the dog. Place hot dog buns on hot grill to toast. Place cheeseburger dog in bun and top with your favorite fixings.