Shape the lamb into 4 patties. Season with salt and pepper.
Heat a skillet over medium heat. When hot, add the patties and cook, flipping once, until seared on each side and cooked through, about 8 minutes.
Sprinkle the burgers with za'atar and serve on a bun with feta, tapenade, tomatoes, arugula, and mint. Serve with lemon wedges.