1. Rinse and pat chicken dry.
2. Bring 4 cups water, 1/4 cup vinegar, the ginger slices, and 2 tablespoons soy sauce to boil in a large saucepan. Add chicken and simmer 1 minute. Remove pan from heat and let stand, covered, until chicken is cooked through, about 5 minutes. Drain chicken and discard ginger slices. Transfer chicken to a cutting board and cut diagonally crosswise into thin slices.
3. Whisk remaining 1/4 cup vinegar, 1 tablespoon grated ginger, 1 teaspoon soy sauce, the sesame oil, vegetable oil, 2 tablespoons water, and the scallion together in a large bowl; salt and pepper to taste. Stir in cabbage, bell pepper, carrot, three-quarters of the chicken slices, and 1 tablespoon sesame seeds until well combined. Mound salad on serving plates, top with remaining chicken slices and sprinkle with remaining sesame seeds. Makes 4 to 6 servings.