Whisk vinegar, 3 tbsp soy sauce, hoisin sauce, ginger, and sesame oil together in medium bowl. Place chicken in single layer in Dutch oven. Pour 1/3 cup vinegar mixture over chicken breast, reserve remaining vinegar mixture to use as dressing. Add remaining 1/3 cup soy sauce and 3 cups water to pot. Bring to boil over high heat. Cover, reduce heat to low, and simmer chicken until cooked through, 7 to 10 minutes.
Transfer chicken to plate, cover loosely with plastic wrap, and refrigerate until cool enough to handle. Shred chicken into long thin strands using two forks.
Transfer chicken to large serving bowl, toss with 2 tbsp of reserved vinegar mixture and season with salt and pepper. Add cabbage, bell pepper, scallions, and remaining vinegar mixture and toss to combine. Sprinkle with chow mein noodles and serve.