Preheat your grill to medium heat. Place the first 6 ingredients in your food processor and pulse until coarsely chopped. With machine running, slowly pour in the 1/2 cup oil and process until a smooth sauce forms. Transfer the sauce to a serving bowl.
Brush the pork with the 1 Tbsp. oil; season with salt and pepper.
Grill the pork, covered, turning occasionally, 12 to 15 minutes, or until just cooked through. Set the pork on a plate, tent with foil and rest 5 minutes. Cut the pork into 1/2" slices and serve with the sauce.