To prepare the artichokes. Snap off the outer dark green leaves of the artichoke. You are looking for the lighter colored inner leaves. Trim the stem flush with the base.
With a serrated knife, cut the cone of leaves at the base to expose the fuzzy choke. Use a paring knife to trim away all the dark green skin, exposing the white flesh underneath.
Use a spoon to scoop out the choke.
With a mandolin or sharp knife, carefully shave the hearts paper thin. Place the shavings in a small bowl with 1 tablespoon of olive oil, the lemon juice, a pinch of salt and a grind or two of pepper. Cover the bowl and place it in the refrigerator.
To make the parmesan crisps. Combine the flour and parmesan in a small bowl by gently rubbing them together through your fingers. Heat a dry non-stick saute pan, sprinkling 1/4 of the parmesan mixture into the center of the pan, keeping the cheese close together in a more or less single layer. When the cheese starts to bubble, remove the pan from the heat and gently work a spatula under the cheese, remove from the pan and place the hot cheese over a bowl or bottle to give it some shape. Repeat for the rest of parmesan crisps. Allow to cool,
For the dressing, whisk the lemon juice, salt and fresh ground black pepper together. Then drizzle in the olive oil while continuing to whisk.
Toss the arugula with the dressing, adding the shaved artichokes. Mound in the center of a chilled salad dish and top with the parmesan crisp and serve.