- Heat the oil in a medium skillet to medium heat. Add the onions and cook 30-40 minutes, stirring occasionally, until they’re browned and caramelized. Season with salt and black pepper. Turn the heat up to medium-high and add the vinegar. Stir until the mixture is thickened. Remove from the heat.
- Heat a grill or grill pan to medium heat. Form the ground beef into 4 equal-sized patties. Season both sides with salt and black pepper. Place the patties on the grill and cook 4-5 minutes per side. Place a slice of cheese on top of each burger and cook 1-2 more minutes or until melted. The burgers are done when a thermometer inserted into the center reaches 160 degrees F.
- Place the burger patties on the bottom bun. Spread mayonnaise on each top bun (if using). Top the burger patties with caramelized onions and arugula and place the top bun onto each burger. Serve immediately.