At least 1 hour before you plan to serve the salad, whisk together the olive oil, lemon zest, juice of 1 ½ lemons, shallots, celery seed, celery salt, anchovy paste, 2 teaspoons salt, and 1 teaspoon pepper.
Then, place the celery in a large bowl and toss it with the remaining juice of the leftover ½ lemon and ½ teaspoon salt (Ina says this is an important step and I must say – after making this, I 100% agree!). Add enough of the dressing in step one to the celery to moisten well. Cover and refrigerate for at least an hour to allow the celery to crisp and the flavors to develop.
When ready to serve, arrange the celery on a platter, shave the Parmesan onto the celery with a vegetable peeler, then sprinkle with walnuts, cilantro leaves, salt, and pepper and serve immediately.