In a blender or food processor, combine the basil, vinegar, olive oil, mustard, pepper, and salt. Blend this until very smooth.
Place the chicken in a ziptop bag or a dish with a lid. Pour the marinade over the chicken and seal the bag. Massage to coat the chicken pieces. Place the bag on a tray to catch any drips and chill in the fridge for 2 to 4 hours.
Heat an outdoor grill or indoor grill pan. Remove the chicken from the marinade and discard the marinade. Cook the chicken on the hot grill until the chicken is cooked through and reaches an internal temperature of 165 degrees. Serve hot, warm, or cold.
Leftovers can be kept in an airtight container in the fridge for up to 4 days. Or freeze for up to 2 months.