Whisk together the vinegar, peanut butter, ginger, chipotle pepper puree, soy sauce, honey, sesame oil, and canola oil in a medium bowl. Season with salt and pepper, to taste. Combine cabbage, lettuce, carrots, snow peas, cilantro, and green onion in a large bowl. Add the dressing and toss to combine.
Transfer to a serving platter and top with the shredded chicken, chopped peanuts, and mint. Drizzle with chili oil, if desired. Garnish with grilled lime halves.