Prepare the grill. Shape meat into 4 patties and season with salt and pepper. Slice the onion into thick round slices (so they don’t fall through the grill). Take the leaves off the rosemary spring, chop and add to the olive oil. Brush onions with half the olive oil and season with salt and pepper. Grill the onion slices, carefully turning them once, until they are golden and soft. Set aside. Grill meat until desired doneness, about 4–6 minutes per side over a medium-hot fire.

Brush buns or rolls with remaining oil then grill until just toasted. Place burgers on buns, top with onion slices and serve with lettuce, tomato slices and barbeque sauce.