In a medium bowl, combine the chicken pieces and 1/2 cup pesto. Cover and refrigerate for at least 1 hour or overnight. If using wooden skewers, place in water to soak for at least 30 minutes. Place mozzarella balls in a small bowl and toss with remaining 1 tablespoon pesto. Refrigerate until ready to serve. Let both the chicken and the mozzarella stand at room temperature 20 minutes prior to grilling.