In a 4- to 6-quart slow cooker, combine the soy sauce, sugar, vinegar, garlic, ginger, and red pepper. Add the short ribs and carrots and arrange in a single layer. Lay the cabbage on top.
Cook, covered, on high for 5 to 6 hours (or on low for 7 to 8 hours―see note below), until the meat is tender and easily pulls away from the bone.
Transfer the cabbage, short ribs, and carrots to plates. With a large spoon or ladle, skim the fat from the cooking liquid and discard.
If the slow cooker is on the low setting, turn it to high. In a small bowl, whisk together the cornstarch with 1 tablespoon of water until smooth. Whisk into the cooking liquid and cook until thickened, 2 to 3 minutes. Stir in the sesame oil. Spoon the sauce over the short ribs and vegetables and sprinkle with the scallions (if using).