In a small bowl combine soy sauce, lime juice, brown sugar, ginger, garlic, cornstarch, and chili-garlic sauce; set aside.
In a large skillet heat oil over medium heat. Add chicken; cook and stir for 8 to 10 minutes or until no longer pink. Stir soy mixture; add to chicken in skillet. Cook and stir until thickened and bubbly.
Divide chicken mixture and kimchi among warm tortillas. Top with cilantro and, if desired, serve with Korean barbecue sauce.