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For the dressing: Put rice wine vinegar, sesame oil, sugar, sesame seeds, and peanut oil in a jar. Shake well, until the sugar is dissolved. If you don’t have a jar handy, just whisk the dressing ingredients in a medium bowl until the sugar is dissolved.
For the salad: Preheat the oven to 475°F.
Rub each chicken breast with 1 tablespoon olive oil. Season with the salt and pepper. Place the breasts on a baking sheet and bake for 17 minutes. Remove from the oven and let rest for 5 minutes. Remove the skin and bones and discard them. Shred the chicken.
While the chicken bakes, you can get started on the rest of the salad. In a 2-quart sauce pot, heat the four cups peanut oil on medium-high to 325°F. Add the wonton strips and stir with a slotted spoon for 1 minute, or until lightly browned. Remove the strips from the oil and drain on a paper towel covered plate.
Combine the lettuce, cabbage, onion, cilantro, and pickled ginger in a large bowl. Add the shredded chicken and about ¾ of the wonton strips.
Pour the dressing over the salad and toss well. Garnish with the rest of the wonton strips and serve.
To put a fun twist on this meal, I like to use chopsticks, and serve the salad in Chinese take-out boxes. You can find the boxes at most party supply stores.