Shred the chicken into bite-sized pieces. Place the chicken, cucumber, carrots, and bell peppers in a large bowl.
Make the dressing by adding the sugar, ginger, lime juice, soy sauce, red pepper flakes, and garlic in a blender until smooth. Add the peanut butter and water; process until smooth. Add the peanut dressing to the chicken, stir in the cilantro, and mix well.
To assemble, place a large lettuce leaf on each warm tortilla and fill with the chicken mixture. Roll it up and enjoy.