Prepare a fire for high heat and set a grate over it. Pull out a pot large enough to hold all the artichokes, such as a caldero or Dutch oven (or a stockpot if cooking indoors). Fill the pot about halfway with water, and season with salt. Set the pot on the grate (or on the stovetop over high heat) and bring to a boil.
Keeping the stems attached, pull off the toughest outer leaves of the artichokes. Lay them down on their sides and, with a sharp serrated knife (bread knife), slice the spiky top half straight off, then snip off any remaining spiky tips with kitchen shears. Pare off the tough skin around the bottoms and stems, rubbing the cut areas with lemon juice as you go.
Add the artichokes and squeezed lemons to the boiling water, partially cover the pot, and cook over medium heat for about 15 minutes, depending on the artichokes’ size, until they are tender all the way through when pierced with a skewer. Drain the artichokes thoroughly in a colander, stems pointing up. Discard the lemons.
Meanwhile, fill a saucepan with enough water to cover the eggs and bring to a boil over medium heat. Lower the eggs into the water and boil for 6 minutes. While the eggs are cooking, fill a bowl with ice and water. With a wide spider or skimmer, transfer the eggs directly into the ice water to stop the cooking. Crack the shells and peel the eggs, put them in a bowl, and mash roughly with a fork. Set aside.
Carefully cut the artichokes in half lengthwise all the way through the stem. Lay them down on a flat surface and scrape out the fuzzy inedible chokes with a teaspoon. As they are done, arrange them cut-side-up on a wide platter.
Season the mashed egg to taste with a few drops of vinegar, add the parsley, and stir in the aioli. Spoon the mixture over the artichoke hearts, season with fleur de sel, and serve.