Fill a large stockpot two-thirds with water. Add the juice of half a lemon and 1 tablespoon salt. Keep the lemon half to rub on the artichokes. Set the pot over medium-high heat.

Fill a large bowl with water and squeeze the other half lemon into it. Keep the half lemon to rub on the artichokes.

Stack the mushroom slices and cut them into thin, matchstick size pieces.

Make the dressing: Squeeze 2 teaspoons lemon juice into a small bowl. Add the vinegar, then whisk in a pinch of salt, the garlic and the heavy cream. Season with pepper and more salt if necessary. Add the mushrooms, turn very gently, and reserve.

Prepare the artichokes. Break off the stems and pull off a the bottom row of leaves, close to the stem, if you can. Using a very sharp knife, and holding it against the side of the artichoke, cut off all the large leaves from the artichoke, leaving a soft cone of small leaves in the center. Cut off this soft cone of leaves, level with the artichoke heart. Scoop out any choke in the center, using a stainless steel spoon, and rub the artichoke heart all over with the lemon, to minimize discoloration. Trim the base of the heart into an even round shape. Rub again with lemon, and put the artichoke into the bowl of acidulated water.

When the water comes to a boil, cook the artichoke hearts until they are tender through but not mushy, 15-20 minutes. Test them after 15 minutes with a sharp knife.

When the artichokes are cool, cut them into thin slices. Arrange the slices in the center of eight plates, in a fan pattern. Drizzle them with 1/4 teaspoon lemon juice, then drizzle each with ½ teaspoon walnut oil. Drizzle each with salt and pepper.

Mince the parsley leaves. Fold them into the mushrooms, then divide the mushrooms evenly among the plates, atop the artichoke slices. Season with fleur de sel and serve.