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When the water boils, put in the artichokes and boil until a bottom leaf pulls off easily and the base of the artichokes are tender to a fork, about 40 minutes. Drain. Pull the inner core from the artichokes and clean out all the hair-like bits, scraping the cavity clean with a spoon.
For the hollandaise: Melt the butter slowly in a saucepan, leaving it on the heat until it has turned brown (not black). Skim off any foam that rises to the top, and pour the clear butter into a bowl, leaving behind the milk solids. Wash the saucepan.
Put the yolks, 1/4 cup/60 ml water and a pinch of salt into the clean pan. Whisk over medium heat until ribbony, 3 to 5 minutes. If the pan gets too hot, the eggs will scramble, so lift the saucepan on and off the heat occasionally to keep the temperature in check. Take the pot off the heat and slowly whisk in the clarified butter, adding it in a thin stream. Season with salt, pepper, and a few squirts of lemon. It will be thick, but still pour off a spoon. It should be globby, like mayonnaise. Spoon the sauce into the cavity of the artichokes. Eat, by peeling off the leaves and dipping them in the sauce. When you get to the heart, eat it with a knife and fork.