Whisk together the oil, balsamic vinegar, Italian seasoning, garlic, mustard, salt and pepper.
Place the chicken in a separate bowl or ziplock bag. Cover with the marinade.
Add the Pesto to the chicken and coat well.
Allow the chicken to marinate in the refrigerator for at least 3 hours. Overnight is best.
Remove the chicken and let it come to room temperature.
Grill the chicken. Be sure to cook it evening on all sides. You want the internal temperature to be 160° F, and the chicken to be brown on all sides.
Allow the chicken to rest for 10 minutes before serving.
Serve warm with an extra dollop of pesto on top if desired.