Put the two least attractive of the pared artichokes (they will be puréed) to cook 10 minutes before adding the others so that, when the others are barely done, all may be left together to cool in their cooking liquid. Remove the chokes and stew the 6 firm bottoms gently in butter, a dab of butter added to the inside of each, covered, in stainless steel or a large, low earthenware vessel. Keep the flame very low and turn them a couple of times over a period of about 20 minutes to ½ hour. Set aside.
Chop or pass the mushrooms through the medium blade of a Mouli-juliènne and toss them, over a high flame, in
Purée the two “unattractive,” well-cooked bottoms through a fine stainless-steel sieve.
After peeling the eggs, choose the point at which the yolk, barely visible, has settled closest to the wall of the white (only absolutely fresh eggs retain the yolk in the center) and slice off, lengthwise, about one third of the egg, removing the thinnest section of the wall. Chop the white parts removed and add them to the artichoke purée. Sieve the yolks. Season and mash the egg and artichoke purées together, adding the brown butter in which the other artichoke bottoms have cooked. Taste for salt and fill the boat-shaped lower sections of the whites with the purée, molding it firmly, well mounded, with a teaspoon, into the original form of the egg and place each egg in an artichoke bottom, pressing it gently into the bed of mushroom juliènne. Arrange them in a lightly buttered gratin dish.
Stir about two thirds of the cheese into the hot béchamel, spoon it regularly over the eggs, coating them evenly, sprinkle the surface with the remaining cheese, and bake in a hot oven (400° to 425°) for 20 minutes, or until the sauce is bubbling and the surface well colored.