Heat olive oil and 1 tablespoon of butter in a skillet over medium heat. Add the sliced onions, season with a pinch of salt, and cook, stirring occasionally, for about 20 minutes until the onions are golden brown and caramelized. Add balsamic vinegar (if using) and cook for another 1-2 minutes. Set aside.

Butter one side of each bread slice. On the unbuttered side, layer 1 slice of cheddar, a portion of the caramelized onions, a few apple slices, and another slice of cheddar. Top with the remaining bread, buttered side facing out.

Heat a skillet over medium heat. Place the sandwiches in the skillet and cook for 3-4 minutes on each side, pressing gently with a spatula, until the bread is golden brown and the cheese is melted.

Slice in half and enjoy warm!