Melt butter in a large skillet over medium heat.
Add onions and cook for 12-15 minutes, stirring every few minutes. If they start to burn, lower the heat.
Once the onions are soft and golden, add in balsamic vinegar, brown sugar, and a pinch of salt. Cook until the sauce has reduced to a thick syrup, about 3 minutes. Transfer the onions to a bowl, and set aside while you make the burger
Add vinegar and sugar
Form 4 ½” patties with the ground beef. Season them with salt and pepper.
Heat a clean skillet on medium heat. Add your patties and cover the pan. Cook for 5 minutes on each side, or until the meat has cooked all the way through. When there are 2 minutes left add cheese to the burgers.
Place the patty on a toasted bun. Top with balsamic onions, arugula, and mayo.