Roughly cut the cabbage leaves into big pieces. Cut the onion. Add the cabbage, onion, garlic, and ginger to the slow cooker. Add water to fill up to ⅔ of the slow cooker. Add salt and teaspoon pepper.
Cover, and cook for about 4 hours on high or 6 hours on low. Cut the scallions into about 2-inch length pieces, and drop into the soup, with a few minutes remaining. Add salt and pepper to taste.
Remove the excess fat. You can spoon off the fat, or use a fat separator. You can transfer the chicken and vegetables to another pot, and strain the broth if you like cleaner broth.