Combine all ingredients (except olive oil) in the food processor. Pulse while slowly streaming in olive oil until smooth.
Butterfly chicken breasts and add 1-2 Tbsp pesto per breast, season with additional s&p and extra olive oil (if desired). Grill over medium heat (7 minutes first side, 5 minutes second side). Freeze remaining pesto in ice cube tray for later use. Great with pasta, combined with mayo on sandwiches, in soups, etc.