Preheat a skillet over medium heat and add vegetable oil.
Season the chicken breasts with salt and black pepper on both sides.
Cook the chicken breasts in the skillet for 5 minutes per side or until fully cooked and juices run clear.
Remove the chicken from the skillet and let it rest for 5 minutes, then slice into thin strips.
In a large salad bowl, combine shredded Napa cabbage, shredded red cabbage, shredded carrots, sliced red bell pepper, sliced green onions, and chopped cilantro.
In a small bowl, whisk together rice vinegar, soy sauce, sesame oil, honey, grated fresh ginger, and minced garlic until well combined to make the dressing.
Pour the dressing over the salad mixture and toss well to coat all ingredients evenly.
Add the sliced chicken, toasted almonds, chow mein noodles, and toasted sesame seeds to the salad bowl.
Gently toss the salad again to distribute the chicken and crunchy toppings.
Serve immediately or chill in the refrigerator for up to 1 hour before serving for a cooler salad.