Cook the whole chicken in a pot with the carrot, onion and bouquet garni in 1 liter of water (or more, as long as your chicken is covered with water).
Cook for 1h to 1h15 at a gentle simmer. You can use a colander spoon to clean up any impurities that may come to the surface. The chicken is well done when you can see that the meat in the thighs comes away easily from the bones. Remove from heat. Strain the stock through a chinois and leave to cool.
Pan-fry the quartered mushrooms. They should color slightly.
To make the béchamel sauce, melt the butter in a heavy-bottomed saucepan, then add the flour. As soon as the mixture thickens, add the stock, stir well with a whisk, add the white wine and milk, and reduce the sauce for 10 minutes.
Bone the chicken, and cut the meat and dumplings into small cubes. Add them to the sauce with the mushrooms.
Season with salt, pepper and nutmeg.
At the last minute, add the cream and reheat without boiling.
Cook the Alsace pasta for 9 minutes, at a gentle simmer.
Top the puff pastry shells with this filling and serve the rest alongside with Alsace Nüdles pasta.