To make the wontons, heat approximately 3 inches of oil over medium-high heat in a medium skillet or saucepan.  When the oil reaches 350°, add a handful of wonton strips, ensuring not to overcrowd the oil.  Fry until golden and crisp, about 20 to 30 seconds.  Using a slotted spoon, remove the strips to a paper towel-lined plate and sprinkle immediately with salt.  Repeat with the remaining wonton strips.

Preheat the oven to 400°.  Line a small baking sheet with aluminum foil and place the chicken breasts on the pan.  Drizzle with olive oil and sprinkle with salt and pepper.  Roast for 30 to 35 minutes, until the chicken is cooked through.  When cool enough to handle, remove the skin and cut the chicken from the bone.  Shred the meat into bite-sized pieces.

In the blender, combine all of the dressing ingredients and blend until smooth.

Using the shredding disk in a food processor, shred the Napa and red cabbages.  Alternatively, you can use the coarse side of a box grater or a knife to cut into thin slices.  Add the cabbage to a large bowl, add the cilantro, carrots, red onion, scallions and chicken. Pour in about 1/2 of the dressing and toss to coat.  Top with the crispy wontons.  Serve the remaining dressing on the side.