In a small saucepan, bring 1/2 cup water to a boil with the vinegar, sugar, 1 teaspoon soy sauce, 2 teaspoons garlic, the crushed red pepper and 1/2 teaspoon salt. Simmer until the sauce is reduced to 1/2 cup, about 5 minutes. Let cool.
Preheat the broiler. In a large bowl, combine the pork, cilantro, ginger, remaining 3 teaspoons soy sauce and 1 teaspoon garlic. Form 8 oval patties about 1/2-inch thick and broil, turning once, until cooked through, about 4 minutes per side. Spoon into lettuce cups and top with the carrots. Serve with the dipping sauce.