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Melt 1 Tbsp. butter with the olive oil in a large skillet over medium low heat. Add the onions and cook, stirring occasionally, until soft and just beginning to caramelize.
Lay out 2 slices of bread. Top with half the cheese, half the ham, then the onions, then the remaining ham, then the remaining cheese and top with the remaining bread slices.
Using the same skillet and onion drippings, add and melt the remaining butter over medium low heat. Place both sandwiches in the skillet and press to slightly soak up onion dripping and butter mixture and flip the sandwich.
Cook, pressing gently with a spatula, until the bottom is golden and the cheese begins to melt, about 4 minutes, and then flip the sandwich again and cook, pressing gently with a spatula, until bottom is golden, about 4 more minutes.
Spray skillet with cooking spray. Heat skillet over medium heat for 1 minute. Add onions and cook until they begin to soften. Add vinegar and cook 2 minutes or until most of the liquid has evaporated.
Top 2 slices bread with ham, cheese and onion mixture and remaining bread slices.
Spray skillet with cooking spray add sandwiches to pan cook until browned and cheese is melted.