For the artichokes: Fill a pot (large enough to hold both the artichokes) with 2 inches of water. Add salt to taste and the parsley sprigs and lemon slices if using.
Trim each artichoke by holding it firmly on its side and slicing off the top 1/2 inch or so, then cutting off the stem close to the base. Remove the smaller tough leaves around the base of the artichoke. Rinse the artichoke, separating the leaves as you do this. If you wish, you can cut off the tips of each leaf with scissors.
Put the artichokes in the pot (the stem ends can be up or down, it doesn't matter), cover the pot and bring the water to a simmer over high heat. Reduce the heat to medium and continue to simmer until a knife slides easily into the base of an artichoke or a leaf comes out very easily when pulled, 25 to 40 minutes. Make sure the water doesn't evaporate; check after about 20 minutes and add more if needed.
For the dipping sauce: Meanwhile, in a small bowl, blend together the mayonnaise, parsley, lemon juice and harissa. Season with salt and pepper.
Serve the artichokes stem-side down on plates, making sure there is room on the plates to discard the leaves (or place a communal bowl in the middle of the table for people to toss in used leaves). Serve the sauce in tiny bowls on the side.
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