Heat the oil in a pot over medium-low. Add the onions and cook, stirring, until very tender and sticky, about 10 minutes. Remove from the heat. Grill the burgers over medium to medium-high until entirely cooked through, about 4 minutes per side. Top with the cheese and onions; cook 1 minute more. Spread the buns with mayonnaise; top the bottom bun with lettuce and tomatoes. Set on the burgers and top buns and serve.