This spring vegetable makes a simple and delicious appetizer or side dish.
Trim 1 inch from artichoke stems and a quarter inch from tops. Remove tough outer leaves. Using scissors, cut sharp tips off leaves. Fill a large saucepan with 5 quarts cold water. Squeeze the juice of 2 lemons into water and add lemon halves to saucepan. Add artichokes, 3 teaspoons salt, and 2 tablespoons olive oil. Place a smaller lid on top to keep artichokes submerged and bring to a boil. Gently boil over medium-high heat until artichokes are tender when pierced in the heart with a paring knife, 25 to 30 minutes. Drain and let cool.
In a small bowl, combine juice from remaining lemon (2 to 3 tablespoons), vinegar, mustard, remaining salt, and pepper. Whisk in remaining olive oil, shallots, olives, and 2 tablespoons Parmesan.
Cut stems of artichokes flush to the base and serve on the side, if desired. Spoon dressing evenly over artichokes. Sprinkle with remaining Parmesan and serve.