Make the lemon za'atar dipping sauce: Put the mayonnaise, lemon juice, sesame seeds, thyme, mustard, sumac, salt, and cayenne in a small mixing bowl. Stir well with a rubber spatula or a spoon to combine. (Note: The sauce can be refrigerated, with plastic wrap pressed against the surface, for up to 2 days.).
Prepare the artichokes: first cut off the bottom portion of the stem, leaving only 1 inch of the stem attached. Using a sharp knife (sometimes a serrated knife works best for this), cut off the tip of the artichoke, slicing straight across about 2 inches down. Using scissors, clip off the sharp spikes on the tips of the remaining leaves.
Put the artichokes in a large pot, and add enough water to reach three-quarters of the way up the artichokes. Squeeze the lemon quarters into the water and then throw the quarters in as well. Add the salt and olive oil to the water. Cover, and cook over high heat until the water comes to a boil, about 15 minutes. Reduce the heat to low and simmer, still covered, for 30 minutes. Test an artichoke to see if it's done by gently tugging one of the lower leaves. It should come off easily but with resistance, and the whole leaf should pull out in one piece, not fall apart.
Turn off the heat. Remove the artichokes, put them upside down on a plate to drain, and let them cool to room temperature, 15 to 20 minutes.
Once they have cooled, cut each artichoke in half lengthwise. With a spoon, gently scoop out and discard the fuzzy inner choke and the pointy purple leaves that line the inside. Cut each half in half again, creating small portions that are more easily shared. Arrange them on a platter and serve with the lemon za'atar dipping sauce.