Have your butcher cut the turkey into 4 pieces and remove the bones from the breast. (Save the neck and backbone for the gravy.) Lay the pieces out on a platter and drizzle with extra-virgin olive oil and freshly squeezed lemon juice. Season well with salt and freshly ground black pepper. Cover and refrigerate until ready to cook.
Now prepare the pesto. Toast the walnuts in a hot oven until golden brown. When done season immediately with salt and then set aside to cool. Toast pine-nuts in a saute pan over medium heat (if not already toasted) and set aside. Put all the ingredients into a food processor and blend. Drizzle some extra-virgin olive oil into the blender to achieve a smooth consistency, but still has texture. Season with salt and pepper and set aside.
To cook the turkey, heat the grill to medium and wipe the grate with some olive oil. Put the turkey on the grill, skin side down, and cook for 30 minutes. Turn the bird over and continue grilling, basting with more olive oil for another 30 minutes. Test the internal temperature of the thigh with a meat thermometer. Turkey is done when thermometer reads 180 degrees F.
Set aside, cover with foil, and let it rest for about 10 minutes before carving. Serve with a healthy coating of celery-walnut pesto over the top.