Heat oil in a large skillet over medium heat. Add the onions, sugar and salt and pepper to taste. Cook, stirring frequently for about 20-25 minutes or until golden in color and wilted.
Form the ground beef into 4 equal patties. Season on both sides with salt and pepper. Heat a griddle or large skillet to medium-high heat. Add the patties and cook, turning frequently until the burgers are fully cooked (the internal temperature has reached 160 degrees F.). During the last few minutes of cooking top each patty with a slice of white cheddar cheese. Cook until the cheese has melted.
Place each burger on a bun and top with some of the caramelized onions.