Preheat oven to 350 degrees F. For the Celery-Walnut Pesto, arrange the walnuts and pine nuts on separate ends of a baking sheet and toast for 8 to 10 minutes, until golden brown, stirring once. Season walnuts immediately with salt, then set both nuts aside to cool. Combine cooled walnuts and pine nuts, garlic, parsley, celery leaves, lemon juice, and Parmigiano-Reggiano in a food processor; pulse to combine. Drizzle the 3/4 cup oil into the processer and blend until the mixture is combined but still has texture. Season with salt and pepper.

Drizzle the 4 turkey pieces lightly with oil and lemon juice. Season with salt and pepper. Preheat the grill to medium. Quickly and carefully wipe the grate with oiled paper towels to create a nonstick surface. Put the turkey pieces on the grill, skin sides down, and cook, uncovered, for 30 minutes. Turn the pieces over and baste with more oil. Continue grilling about 30 minutes more, until the juices run clear (180 degrees F in the thigh). Set aside, cover with foil, and let stand about 10 minutes.