In a small bowl, whisk together the peanut butter, soy sauce, rice vinegar, honey, and warm water until completely smooth.
In a medium bowl, toss the shredded cooked chicken with half of the prepared dressing until evenly coated.
Lay the four large tortillas flat on a clean surface.
Divide the sauced meat, carrots, bell pepper, cabbage, and cilantro evenly among the center of each tortilla.
Fold the left and right sides of each tortilla inward, then roll tightly from the bottom to the top. Serve immediately with the remaining dressing for dipping.