Heat butter and 1 tablespoon olive oil in a skillet. Add the onions and cook over medium-low heat for 15-20 minutes until caramelized. Add balsamic vinegar, stir, and cook for another 2 minutes. Set aside.

Season the steak with salt and black pepper. Cook in a skillet or grill over medium-high heat for 2-3 minutes on each side for medium-rare.

Slice the rolls and lightly toast them. Layer steak slices on the bottom half of the roll.

Top with caramelized onions and a slice of cheese. Place in an oven or skillet for 2-3 minutes to melt the cheese.

Close the sandwich with the top half of the roll, slice diagonally, and serve warm.