Whisk all peanut sauce ingredients together, adding warm water gradually until smooth and pourable. Set aside.
Warm tortillas in a dry skillet 20 to 25 seconds per side until pliable.
Spread 2 tbsp peanut sauce across the lower third of each tortilla. Layer chicken, cabbage, carrots, cucumber, and green onion in a tall, compact mound.
Scatter cilantro and chopped peanuts over filling.
Fold sides inward and roll tightly from the bottom. Rest seam-side down 30 seconds before cutting diagonally. Serve with lime and extra sauce.