To make the marinade, combine the first 7 ingredients in a shallow dish.
Thread the chicken strips onto bamboo skewers in a serpentine fashion.
Place the skewers into the soy sauce mixture and let marinate in the refrigerator at least 2 hours, although overnight is preferable.
Make the peanut sauce by combining the next 7 ingredients (peanut butter through garlic) in a saucepan. Mix thoroughly then add cayenne pepper.
Cook over moderate heat, stirring constantly, until the sauce is as thick as heavy cream (about 15 minutes).
Transfer to a food processor or blender and puree briefly. Add chicken broth and soymilk and blend until smooth. This mixture can be made several hours ahead and stored in the refrigerator. Bring to room temperature before serving.
Prepare moderate-hot charcoal coals or preheat a broiler.
Cook the skewered chicken, turning several times and basting with the marinade, until crispy on the outside but still moist on the inside, about 8 minutes. Sprinkle grilled chicken with lime zest and garnish with cilantro leaves. Serve with the peanut sauce for dipping.