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)*, 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).
*Because different slow-cooker models cook at different rates, the total time is presented as a range. Check for doneness at the low end of the range (but not before, or the escaping heat will slow the cooking). If the food is not done, continue cooking to the end of the suggested range.