Place the chicken breasts in a medium saucepan and add enough water just to cover. Add 1 tablespoon of the lime juice, 1 tablespoon of the fish sauce, 1 tablespoon of the rice vinegar, and the reserved cilantro stems and bring just to a boil. Reduce the heat to a simmer and cook, partially covered, for 10 minutes. Remove the pan from the heat and allow the chicken to sit in the poaching liquid for 45 minutes. Remove the chicken from the liquid and allow it to cool completely. Transfer the chicken to an airtight container and refrigerate until thoroughly chilled, at least 2 hours and up to overnight.
In a non-reactive bowl combine the remaining lime juice, fish sauce, rice vinegar, the sugar, chiles, garlic, red pepper, and wasabi, if using, and stir until the sugar is dissolved.
In a large bowl combine the cilantro leaves, onion, cabbage, carrots, mint leaves, and Thai basil leaves and toss to combine. Add 1/3 of the lime-chile mixture and toss. Let stand for 10 minutes, refrigerated, before serving.
Divide the salad evenly among 4 plates. Thinly slice the chicken breasts against the grain and divide evenly among the top of the salads. Garnish with the scallions, peanuts and crispy shallots and serve immediately, with the remaining lime-chile sauce on the side for dressing salads, to taste.
Preheat the peanut oil to 360 degrees F. Add the sliced shallots, in batches, and fry until golden brown and crispy, 1 to 1 1/2 minutes. Remove the shallots using a slotted spoon or skimmer and transfer to paper towels to drain. Season with salt to taste.