In a large casserole or saucepan, sauté the onion very gently in the butter until soft, making sure that the onion does not brown. Add the carrot and celery and continue sautéing for a minute or two more.

Then add the chicken pieces, turning them in the butter for a few minutes, just until the meat stiffens a bit.

Add enough broth to cover, nestle the bouquet garni among the chicken pieces, and simmer, covered, until the chicken meat is done. The cooking time will vary according to the size and type of meat. Twenty or 30 minutes will do for chicken tenders (boneless, skinned chicken thighs) while chicken breasts, for example, will take much less time, probably no more than 10-15 minutes at the most. Add the potato cubes to cook along with the chicken and vegetables about 15 minutes before the chicken is done.

Turn of the heat, and remove the chicken pieces and the bouquet garni from the pot. When the chicken has cooled off a bit, cut it into bite-sized pieces. (If using bone-in chicken pieces, bone them before cutting up the meat.) Discard the bouquet garni.

Meanwhile, whisk together the egg yolk and the cream in a bowl. Then whisk in a bit of the hot broth from the pot to 'temper' the mixture. (In other words, you are raising the temperature of the mixture—this helps prevent the egg from scrambling during the next step.) Then, stirring all the while, add the mixture to the pot.

Bring the pot back to the simmer, stirring constantly, until the broth thickens enough to coat a spoon. The texture should be velvety but not too thick. Be careful not to let the broth come to a boil, or the egg may scramble. Add the chicken pieces back into the pot, folding them gently into the sauce. Allow to simmer very gently for a minute or two to reheat the chicken.

Taste for seasoning and serve in heated bowls, garnished with a sprinkling of minced parsley