Soak the bread in the lukewarm milk, wring out well and turn through a meat grinder. Briefly sauté the onions, parsley and mushrooms in a tsp. of clarified butter. Mix with the bread, whole egg, egg yolk, salt and pepper. Rinse the fish, dry it with kitchen roll and stuff it with the mixture. Sew it up with kitchen thread. Sprinkle with salt and pepper, place in a gratin dish greased with clarified butter and baste with the remaining liquid clarified butter. Roast for 15 min. in the heated oven at 220 °C, then baste with sweet whipped cream and finish stewing for another 20 min. or so.
Reduce oven temperature to 180 °C. 10 min before the end of the braising time, pour off the resulting liquid into a small saucepan. Add the juice of one lemon. Bring to a boil, then remove from heat. Stir the remaining 2 egg yolks and add to the sauce, stirring until just before boiling point.
Serve the carp in the gratin dish. Add the other lemon cut into quarters. Serve the sauce separately.
By the way: Did you know that clarified butter is obtained by extracting water, milk protein and lactose from butter?