Place the chicken breasts in a medium saucepan and cover with water. Add salt and pepper. Bring to a boil over medium-high heat, then reduce heat to low and simmer for 8 to 10 minutes or until the chicken is cooked through.
Remove the chicken from the water and let cool. Once cooled, shred the chicken using two forks.
In a large salad bowl, combine the shredded Napa cabbage, shredded romaine lettuce, shredded carrots, sliced red bell pepper, and sliced green onions.
Add the shredded chicken, toasted sliced almonds, chopped cilantro, and chow mein noodles to the salad bowl.
In a small bowl, whisk together the rice vinegar, soy sauce, sesame oil, honey, grated fresh ginger, minced garlic, and vegetable oil until well combined.
Pour the dressing over the salad and toss gently to coat all ingredients evenly.
Serve immediately or refrigerate for up to 1 hour before serving to allow flavors to meld.