Combine meats, mint, oregano, onion, egg and cumin gently with hands. Let rest 45 minutes.
Meanwhile, for the Greek relish, combine all ingredients together and season with salt and pepper to taste. Reserve.
Heat grill to high. Shape into six lamb burger patties and season with salt and pepper. Cook 3 minutes per side, or to desired doneness (130˚F internal temperature for medium-rare). Char lemon on grill, and squeeze over grilled lamb burgers. Serve with warmed pita or naan and relish.