Add butter and oil to a medium skillet.
Add onions and cook over low-medium heat for 30 minutes. Try not to stir the onions too much. The onions should be a dark brown color.
Add parsley to the caramelized onions and set aside.
Butter each slice of bread. Place buttered side down on a large skillet or griddle.
Add 3 extra thin slices of cheese on each side. (6 slices on each sandwich total).
Place onions on one side of bread. When the cheese has melted put sandwiches together and take out of the pan.
Cut in half and serve immediately with a side of hot beef broth if desired.