Prepare caramelized onions according to recipe link. Slice onion rolls and butter outsides of each. Heat skillet over medium heat. Place bottom half of each roll into skillet. Pile half of onions on each, topped with half of the cheese. Replace top halves of buns.

Grill until bottoms are golden brown. Carefully turn over each sandwich and cook until tops are golden.

Serve with warm beef broth for dipping.