Squeeze 1 lemon into a large bowl and add enough cool water to fill about halfway. Drop squeezed lemon halves into bowl.
Using a mandolin or sharp chef’s knife, slice fennel as thinly as possible and place in a separate large bowl. Thinly slice celery on a diagonal and add to bowl with fennel. Squeeze remaining lemon over the fennel and celery.
Working one at a time, trim the bottom off each artichoke stem. Pull off the tough green outer leaves until you reach the fine, thin pale yellow leaves. Slice off the fine, yellow leaves at the top, to the spot where they meet the heart. Trim off the tough skin at the top of the heart; peel stem. Using a spoon or a small ice cream scoop, remove the choke (the furry part in the center of the heart). Place the artichoke hearts in the bowl with lemon water.
Remove artichoke hearts from the lemon water and pat dry. Using a mandolin or sharp knife, slice as thinly as possible and add to the bowl with the fennel mixture. Gently toss all the ingredients, then add 2 tablespoons oil, ¼ teaspoon salt, and ½ teaspoon pepper and gently toss one more time.
Divide artichoke mixture among 4 plates. Using a vegetable peeler, shave Parmesan over each. Top with parsley, sprinkle with flaky sea salt, if desired, and serve.
Photography by Robert Bredvad. Food and Prop Styling by Maeve Sheridan.