Sprinkle chicken with pepper. In a 3- to 4-quart slow cooker, place chicken, celery, onion, and garlic. Combine in a small bowl the hoisin sauce, soy sauce, ginger, sherry, chili sauce, and sesame oil. Pour over mixture in slow cooker. Stir to coat.
Cover and cook on low-heat setting for 5 to 6 hours or on high-heat setting for 2 1/2 to 3 hours.
Remove chicken from cooker, reserving 1/2 cup of the cooking liquid. When chicken is cool, using two forks, pull chicken apart into shreds.
To make dressing, use a screw-top jar and combine the reserved 1/2 cup cooking liquid and the rice vinegar. Cover and shake until well mixed; set aside.
Combine in a large salad bowl the chicken, romaine, carrots, cashews (if desired), and cilantro. Before serving, shake dressing and drizzle over salad. Toss to coat. Enjoy!