Make the peanut sauce: In a medium bowl, whisk together the peanut butter, soy sauce, honey, rice vinegar, sesame oil, minced garlic, grated ginger, and lime juice. Add warm water one tablespoon at a time if needed to reach desired consistency. Set aside.
Cook the chicken: Heat vegetable oil in a large skillet over medium-high heat. Add chicken pieces in a single layer and cook for 5-6 minutes, stirring occasionally, until golden brown and cooked through (165°F internal temperature).
Season the chicken: Remove chicken from heat and toss with about 1/4 cup of the peanut sauce. Set aside.
Prepare the vegetables: In a large bowl, combine coleslaw mix, shredded carrots, and half of the cilantro and green onions. Drizzle with some peanut sauce and toss to coat lightly.
Warm the tortillas: Heat tortillas in microwave for 15-20 seconds or in a dry skillet for 30 seconds per side until pliable.
Assemble the wraps: Spread a spoonful of peanut sauce on each tortilla. Top with chicken mixture, vegetables, remaining cilantro, green onions, and crushed peanuts. Roll tightly and serve immediately.