Preheat the grill or broiler to high. Combine the bulgur, beef, tomato paste, parsley, the remaining 1/8 teaspoon salt and pepper in a medium bowl; knead thoroughly to combine. Shape into two 3/4-inch-thick patties. Grill or broil on a lightly oiled rack until browned and cooked through, about 5 minutes per side. Place the patties on buns, top with the caramelized onions and garnish with lettuce.