Clean the artichokes by removing all the tough outer leaves until you get to a pale yellow center. Peel the outer tough skin of the artichoke stems with a paring knife. Remove the purple leaves and hairy portion covering the choke; discard. Trim 1/3 off the tops of each artichoke, rub with lemon and finely chop. Squeeze extra lemon juice on artichokes to prevent from browning.
Plate the artichokes and season with olive oil and extra lemon juice, if needed. Then add pepper and salt, to taste. With a vegetable peeler shave the Parmigiano cheese and top the artichokes. Lightly drizzle with more olive oil if needed.