Fill a large bowl with water. Squeeze the juice from the quarters of 1 lemon into the bowl. Working with 1 artichoke at a time, remove the tough outer leaves to expose the light yellow core. Peel the stem end of the artichoke, keeping about 1 inch of the stem intact and exposing the pale, tender core. Trim 1 inch off the top of the artichoke and cut the artichoke in half lengthwise through the center. Using the tip of a spoon, scoop out the furry choke. Add the cleaned artichoke halves to the lemon water. Keep the artichokes in the water until ready to cook; they may be stored in the refrigerator for 1 to 2 days. Just before cooking, spread the artichokes on paper towels, sliced side down, to drain and pat dry.
In a large sauté pan over medium-high heat, warm the olive oil until nearly smoking. Sprinkle the pan generously with salt and pepper. Carefully place the artichokes, sliced side down, in the pan, making sure they lie flat. Season with salt and pepper. Cook, shaking the pan occasionally to keep the artichokes from sticking, until they are evenly browned underneath (about 6 to 10 minutes). Using tongs, lift 1 or 2 artichokes up to check for doneness.
Add the quarters of 1 lemon to the pan, place a piece of foil over the pan and cover with a lid. Reduce the heat to medium-low and cook for 5 minutes more. Remove the pan from the heat and keep covered. Let the artichokes cool to room temperature, 30 to 45 minutes.
Transfer the artichokes to a serving platter. Squeeze the juice from the cooked lemon wedges into the pan and whisk to scrape up any browned bits stuck to the pan bottom. Pour the juice over the artichokes, garnish with the remaining lemon quarters and serve.
Yields 4