Method:
Prepare the grill. Shape sirloin into 4 patties and season all over with salt and pepper; set aside. Remove leaves from rosemary sprig, then roughly chop them and transfer them to a small bowl with the oil. Brush onions with half of the oil mixture, then season with salt and pepper. Grill onions, carefully turning them once, until golden and soft. Transfer to a plate and set aside. Grill patties over a medium hot fire until desired doneness, about 4 to 6 minutes per side. When they are almost done, melt cheese slices on top.
Meanwhile, brush buns with remaining oil, then grill until just toasted. Place burgers on buns, top with onion slices and serve with lettuce, tomatoes and barbeque sauce.