Combine the Napa cabbage, red cabbage, shredded chicken, green onions, cucumbers, carrots, snow peas, bean sprouts and cilantro in a very large mixing bowl.
Using your hands, gently pull the mung bean vermicelli noodles apart into smaller portions. Place some paper towels on 2 dinner plates. In a small skillet add an inch of cooking oil (the smaller the skillet, the less oil you need) and heat the oil until it ripples but does not smoke. Fry one small handful of noodles at a time until they puff up, about 5 seconds, flip over with tongs and fry the other side for a second or two. Remove from oil with tongs, place on the prepared plate and sprinkle with a little salt. Repeat until all the noodles are fried.
Add the dressing to the vegetable/chicken mixture and toss well to coat. Crumble the fried mung bean vermicelli noodles into the salad; add the peanuts, season with about 1 teaspoon of salt and ½ teaspoon of pepper, toss again and serve.