Cook the pasta in salted water until it is not quite al dente.
Soak the porcini mushrooms in boiling water for at least 30 minutes, then squeeze and chop. Keep the liquid. Clean the mushrooms and cut them into fine slices.
Remove the crusts from the white bread and soak it in the milk. Peel and finely chop the garlic clove, chop the parsley. Melt some butter in a pan, sauté the mushrooms and pour the soaking liquid over them. Simmer until it has almost evaporated.
Put the mushrooms in a bowl and let them cool. Separate the eggs, add the yolk to the mushrooms. Squeeze the white bread, shred it and add it to the mushrooms, mix everything well, season with salt and pepper.
Beat the egg whites until stiff and fold them into the mushroom mixture. Butter an ovenproof dish, pour in half of the pasta and spoon the mushroom mixture over it. Finely slice the Gruyezer and Bel Paese and pour over the pasta, shave the Parmesan and pour half over the pasta.
Top with the remaining tagliatelle.
Mix the cream with the remaining Parmesan and spoon over the pasta, adding a few flakes of butter on top. Bake covered with foil in a preheated oven for 15 minutes at 180° degrees, then brown without foil for another 10 minutes at 220° degrees. Let rest for a few minutes before serving to allow the flavor to develop.