Put the lemon juice in a medium bowl and set aside.
Remove each artichoke individually from the lemon water and halve them. Using a teaspoon or melon baller, scoop out and discard the fuzzy inner choke and trim off any rough, pointy bits.
Slice each artichoke half into the thinnest possible wedges, then add them to the bowl with lemon juice and mix well. Slice the trimmed stems into the thinnest possible rounds and add to the bowl. Add the olive oil and salt and pepper to taste and mix well.
Garnish with the mint leaves. Use a vegetable peeler to shave slices of Parmigiano-Reggiano over the salad. Serve immediately.
Begin by filling a large bowl with cold water. Add the juice of 1 lemon and drop in the squeezed lemon halves. Snap off the tough outer leaves of each artichoke just above the base, one at a time. Continue to remove the layers until you reach the light-colored inner leaves. Cut off the stem, leaving about 1 inch attached to the base. Using a small knife with a short, thin blade, or a vegetable peeler, peel off the fibrous outer skin from the removed stem until you reach the pale green inner flesh. Drop it in the bowl with lemon water.
Carefully peel away the tough, dark green skin from the base of the artichoke and its trimmed stem. Remove and discard the upper cone of leaves. Hold in the lemon water to prevent oxidizing until ready to use.