Spray cold grate of gas grill with grilling spray. Preheat grill for medium heat. Meanwhile, heat oil in small saucepan over medium heat. Add garlic and ginger and cook 30 seconds or until fragrant. Add ketchup, soy sauce, 3 tablespoons sugar and 1 tablespoon vinegar. Bring to a boil; reduce heat and simmer 5 minutes, stirring occasionally. Remove from heat; set aside.
Microwave remaining 3 tablespoons vinegar and 1 tablespoon sugar in small microwave-safe bowl on HIGH 30 seconds or until hot. Stir until sugar is dissolved; add cucumber to bowl. Cover until ready to serve.
Grill franks 5 to 7 minutes or until hot, turning occasionally.
Drain pickled cucumbers. Spread lettuce leaves evenly with BBQ sauce; top each with a frank, cucumbers and carrots. Sprinkle with cilantro, green onion; top with kimchee, if desired. Serve immediately.