*The number of egg yolks needed to thicken this fairly light sauce will vary depending on the size of the yolks and how much the sauce is reduced before they are added.
PREPARE the COURT-BOUILLON and let it cool. Include in it, if possible, the trimmings of the fish. Have the fish, ready to cook, in the refrigerator.
If you are using the DUCHESSE POTATOES, prepare them and pipe them around the edge of the platter. Brown them in the oven shortly before you finish cooking the fish.
Cook the mushrooms. Peel the tomatoes, cut them into quarters, and seed and drain them carefully. Simmer them with a good lump of butter and salt and pepper until they are tender but the pieces still keep their shapes. Keep the mushrooms and tomatoes warm, or reheat them at the last minute.
Poach the sole in the court-bouillon. When it is done, arrange it, first trimmed and well drained, on the platter (butter the platter lightly if you have just baked the potato garnish on it, and keep it warm.
Reduce the court-bouillon over a brisk fire -until it thickens slightly. Then strain it and measure out the amount specified for the sauce. In a heavy saucepan, combine the reduced fish stock and the butter, white wine, and vermouth. Continue to boil down over high heat, stirring often, until the liquid is reduced to two thirds its original quantity. Remove from the heat and, with a sauce whisk, beat in quickly the lightly beaten egg yolks. Heat the sauce very gently now, beating constantly with the whisk, and remove from the fire as soon as it thickens.
Mask the fish on the platter with the sauce and surround it with the mushrooms and tomatoes, which must first be carefully drained. Serve immediately, and if there is extra sauce, pass it in a sauceboat.
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La Cuisine Chantraine