Finely chop parsley and cilantro; transfer to bowl. Stir in peanuts, all but 2 tsp of the oil, the vinegar and half each of the salt and pepper. Set aside.
Between plastic wrap, pound chicken with meat mallet or bottom of heavy saucepan to 1/4-inch (5 mm) thickness. Brush with remaining oil; sprinkle with remaining salt and pepper.
Place chicken on greased grill over medium-high heat; close lid and grill, turning once, until chicken is no longer pink inside, about 5 minutes. To serve, spoon sauce over chicken.
