In a medium bowl, whisk together the peanut butter, soy sauce, rice vinegar, lime juice, honey, sesame oil, ginger, garlic, and 3 tablespoons warm water until smooth and creamy. Add additional water as needed to create a pourable sauce.
In a separate bowl, combine the shredded chicken with the soy sauce and lime juice. Add about one-third of the peanut sauce and toss until the chicken is evenly coated.
In another bowl, combine the cabbage, carrots, bell pepper, cucumber, green onions, cilantro, lime juice, and salt. Toss until evenly mixed.
Warm the tortillas briefly in the microwave or a dry skillet until soft and flexible.
Divide the vegetable mixture evenly among the tortillas, leaving a border around the edges.
Top each tortilla with the peanut chicken and sprinkle with the chopped peanuts.
Drizzle with additional peanut sauce.
Fold in the sides of the tortillas and roll tightly into wraps.
Slice in half and serve immediately with the remaining peanut sauce for dipping.