To trim each artichoke, cut the stem flush with the bottom. Using a serrated knife, cut about 1 1/2 inches off the top of the artichoke. Using scissors, snip off the pointed tips of each leaf. Rub the artichoke all over with the lemon half.

Pour water to a depth of 1 inch into a large pot and add the pickling spice. Bring to a boil over high heat. Place a steamer rack in the pot, making sure that the water does not touch the rack, and add the artichokes, stem end up. Cover and adjust the heat to maintain a gentle simmer. Steam until a knife pierces the bottoms easily, 40-45 minutes.Remove the artichokes and let stand stem ends up, on several thicknesses of paper towels for a few minutes.

​Meanwhile, make the aioli: in a small bowl, whisk the egg yolk with the warm water, then whisk in the olive oil drop by drop. Once an emulsion has formed, gradually add the oil a little faster until all of it has been incorporated and a mayonnaise-like sauce has formed. In a mortar, combine 1 garlic clove and the salt and pound to a paste. Whisk the paste into the egg-oil mixture. Whisk in a few drops of lemon juice if desired. Add more pounded garlic to taste.

​Using your fingers, gently spread apart the center artichoke leaves. Pull out the prickly innermost leaves. With a spoon, scrape out the hairy choke. Serve the artichokes hot, warm, or chilled. Spoon the aioli into the center of each artichoke or serve alongside it.