Set up a bowl of acidulated water: break the vitamin C capsules into a large bowl of water and squeeze in the juice from 2 of the lemons. Toss in the 4 halves of the squeezed lemon as well. Artichokes will turn brown (oxidize) before your eyes so you must, must, must set up this bowl of acidulated water first!
Working with 1 artichoke at a time, peel off the tough green outer leaves to reveal the spring green color underneath. Trim off the tough outer layer of the stem and any brown spots. Cut the tip of the choke off. Artichokes have a lot of waste, accept it and move on. Very carefully, using a mandoline, very thin shave the artichokes and put them immediately into the lemon water. The artichokes are best when done at least a couple of hours ahead because they tenderize a little.
To assemble the salad:
Put the artichokes in a salad spinner to remove they excess water. Put the artichokes, arugula and Parmigiano-Reggiano in a large serving bowl. Dress with the juice of the remaining lemon, finishing oil and salt, to taste. The salad should be well-dressed and full flavored but not soggy.