In a medium skillet over medium low heat, combine oil and rosemary. Cook for 2 minutes. Add onions, and cook for 20-30 minutes stirring often until golden brown. Transfer to a small bowl and set aside.
Pre-heat a gas or charcoal grill. Form burgers into patties and coat all sides with pepper and salt. Place burgers on grill over direct heat. Grill for 5-6 minutes per side or until cooked to desired doneness. Just before removing burgers from grill, place 2 slices of cheese on each burger. Transfer burgers to a plate or tray and let rest for 10 minutes.
While burgers are on the grill, prepare apricot mayo by stirring together mayonnaise and apricot preserves in a small bowl until smooth.
Spread 1 tablespoon of apricot mayo on both sides of buns. Place lettuce leaves on bottom buns. Place burgers over lettuce. Spoon reserved onions over burgers, then finish with bun tops. Serve.