In a bowl, whisk together peanut butter, soy sauce, rice vinegar, lime juice, olive oil, garlic, honey, and sriracha. Add water a tablespoon at a time until the dressing is smooth and pourable.
If using raw chicken, grill or poach until cooked through (about 10 minutes), then shred or chop. If using rotisserie chicken, just chop and set aside.
In a large bowl, combine lettuce, cabbage, carrots, bell pepper, green onions, cilantro, and half the peanuts. Add the chicken and toss everything together.
Drizzle half the peanut dressing over the salad and toss again. Add more dressing to taste. Top with remaining peanuts and lime wedges. Serve immediately.