Seed and chop cucumber. In a small bowl stir together cucumber, yogurt, 2 tablespoons mint, and salt and pepper to taste. In a large bowl blend together lamb, cumin, remaining 2 tablespoons mint, and salt and pepper to taste. Form mixture into four 1-inch-thick patties. When forming patties, handle meat as little as possible for a juicier burger.
Prepare grill.
Grill burgers on an oiled rack set 5 to 6 inches over glowing coals about 4 1/2 minutes on each side for well-done. (Alternatively, burgers may be cooked in a hot well-seasoned cast-iron skillet or ridged grill pan.)
Toast pita loaves and split horizontally. Transfer burgers to pitas and top with cucumber mixture, eggplant, and romaine.
