In a small bowl, combine the bulgur with ½ cup warm water; let stand for 30 min, or until the bulgur is tender and the water is absorbed. In a large non-stick skillet, heat oil over low heat. Add onions and sugar; sauté until the onions are tender and dark golden, about 15 min. Stir in ¼ cup water and vinegar. Season with salt and keep warm. Prepare the grill or preheat the broiler. In a medium bowl, combine the bulgur, beef, tomato paste, parsley, ½ teaspoon salt and pepper; mix thoroughly but lightly. Shape into four ¾ inch-thick patties. Grill over high heat or broil on a lightly oiled rack until browned and cooked through, about 5 min per side. Blot the patties with paper towels. Place each patty on a bun, top with the caramelized onions and garnish with lettuce.