In a medium bowl, whisk together peanut butter, 2 tablespoons soy sauce, 1 tablespoon sesame oil, lime juice, minced garlic, grated ginger, and honey until smooth and well combined. Set aside.

Cut the chicken breast into thin strips and place them in a bowl. Add remaining 1 tablespoon soy sauce and 1 tablespoon sesame oil. Mix well and let marinate for 10 minutes while preparing vegetables.

Wash and dry lettuce leaves, shred purple cabbage, and julienne carrots into thin matchsticks. Chop fresh cilantro. Arrange all vegetables in separate bowls for easy assembly.

Heat a large skillet over medium-high heat. Add marinated chicken strips and cook for 6-8 minutes, stirring occasionally, until chicken is golden brown and cooked through. Remove from heat and let cool slightly.

Warm flour tortillas in a dry skillet for 30 seconds on each side, or wrap in a damp paper towel and microwave for 20 seconds until soft and pliable.

Spread 2 tablespoons of peanut sauce down the center of each tortilla. Layer with lettuce, purple cabbage, carrots, cooked chicken, and fresh cilantro. Drizzle with additional peanut sauce if desired.

Fold in the sides of the tortilla and roll tightly from bottom to top. Cut each wrap in half diagonally and serve immediately with extra peanut sauce on the side.