Make the dressing: Add the rice vinegar, soy sauce, olive oil, toasted sesame oil, honey, minced ginger, and garlic to a medium bowl or liquid measuring cup. Whisk until fully combined and the honey is dissolved. Taste and adjust if needed. Set aside.

Build the salad: In a large mixing bowl, combine the shredded napa cabbage, red cabbage, shredded chicken, sliced bell pepper, carrots, almonds, green onions, and cilantro. Toss gently to distribute everything evenly.

Dress and serve: Pour the dressing over the salad a little at a time, tossing as you go, until every ingredient is lightly coated. Start with half the dressing and add more to taste. Serve immediately for best crunch.