The tender leaves and hearts of artichokes are delicious dipped in aioli. If you like, snip the prickly points of the leaves with scissors before you cook the artichokes.
Cut each artichoke lengthwise into quarters, through the stem. Using a spoon, remove the fuzzy choke from the heart of each quarter, and immediately squeeze juice from one of the lemon halves over all artichokes to prevent them from discoloring.
Place all the ingredients, including the remaining lemon half, into a large saucepan; fill with enough cold water to cover artichokes. Cover pan, and bring the water to a simmer over high heat. Reduce heat to medium-low; cook just until artichoke hearts can be pierced with a fork but are not too soft, 10 to 15 minutes. Remove from heat, and let artichokes sit in cooking liquid until ready to serve, up to 10 minutes. Transfer carefully to serving plates with a slotted spoon, allowing excess liquid to drain off.