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 6-7 minutes on each side or until fully cooked and internal temperature reaches 165°F (74°C).
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, honey, toasted sesame oil, grated fresh ginger, minced garlic, and lime juice until well combined.
Pour the dressing over the salad and toss thoroughly to coat all ingredients evenly.
Add the sliced chicken on top of the salad and gently toss again to combine.
Sprinkle toasted sliced almonds and crispy chow mein noodles over the salad before serving.