Preheat grill to medium-high. Bring a large pot of water to a boil. Combine basil, Parmesan, walnuts, garlic, 1/4 tsp. salt, and 1/8 tsp. pepper in a food processor; pulse until finely chopped. Slowly pour 1/4 cup oil through the feed tube; continue processing until the mixture is smooth. Transfer to a medium bowl and set aside.