Step 1In a large bowl, whisk the Dijon mustard, ice cube and lemon juice. Add the olive oil a few drops at a time, whisking to emulsify. Season with salt and pepper. Set aside. (Makes two-thirds cup.)
Step 1Trim the ends of the stems and pare down the artichokes to the tender leaves and hearts. As you work, place the prepared artichokes in a bowl with enough water and the lemon juice to cover. Remove the prepared artichokes from the lemon water and pat dry.
Step 2Using a mandoline, cut the artichokes into very thin, lengthwise slices. Place the slices in a large bowl and toss with 1 to 2 tablespoons of dressing. Still using the mandoline, delicately slice the celery heart and add to the bowl with the artichoke slices. Add one-fourth cup Parmigiano-Reggiano, tossing gently, and season to taste with salt and pepper.
Step 3Place the arugula in another bowl and toss with 1 to 2 tablespoons dressing or to taste. Gently combine the arugula and the artichoke-celery mixture. Add the remaining Parmigiano-Reggiano and transfer to a chilled salad bowl.
Step 4Drizzle the salad with the remaining teaspoon of olive oil. Pass the remaining dressing on the side.