Mix water and cornstarch in a small mixing bowl. Set aside.
Stir together tamari, brown sugar, rice vinegar, chili paste, ginger, sesame oil, and black pepper in a saucepan and bring to a boil. Add water and cornstarch mixture. Cook for 3 – 5 minutes, stirring frequently, until Korean BBQ sauce thickens.
Heat wok on medium heat. Add carrots and blanch for 2-3 minutes. Add some Korean BBQ sauce mixture, stir, and remove from heat.
Heat wok on medium heat. Add red cabbage and blanch for ~1 minute (do not overcook as red cabbage loses its color very quickly). Add some Korean BBQ sauce mixture, stir, and remove from heat.