Cook the whole chicken in a pot with carrot, onion and the bouquet garni in 1 liter of broth. After 60 to 75 minutes cooking, the meat should be tender and come off easily.
Let cool and reserve the filtered broth.
Fry mushrooms cut into cubes.
For the Béchamel sauce, melt the butter in a heavy saucepan, then add the flour. As soon as the mixture thickens, add broth, stir well with a whisk. Add the white wine and milk and let reduce the sauce for 10 minutes. Remove bones from the chicken and cut the flesh and quenelle into small cubes. Add them in the sauce.
Season with salt, pepper and nutmeg.
At the end, add the crème fraîche as well as the beaten egg yolks.
Reheat without boiling. Garnish the vol-au-vent cases and serve the remaining on the side with pasta or “spätzle”.