Trim all visible fat and tendons from chicken breasts, pound until even, and cut each breast in half lengthwise to make two strips.
Combine first seven ingredients in a small bowl and wisk together. Put chicken pieces in ziploc bag, pour in marinade, then seal bag and turn over several times so chicken is well-coated with marinade. Let marinate in refrigerator, a minimum of a few hours, or all day.
To cook chicken, first spray grill with olive oil, or wipe with paper towel dipped in oil if you don't have an olive oil mister. Preheat charcoal or gas grill to medium high.
Thread two chicken pieces onto each skewer, folding back and forth as many times as you can to get a lot of the chicken surface exposed to the grill. Grill chicken about 10 minutes, turning half-way through until it's nicely browned and feels firm (but not hard) to the touch.
For peanut sauce, combine remaining ingredients in a food processor and mix together. Transfer to a small saucepan and heat over low heat.
Serve chicken hot, with some sauce on the side for dipping chicken pieces.