Melt butter to medium-high heat. Sear the chicken pieces on both sides in batches. Sprinkle two third of the salt and all of the pepper onto the meat while frying. To get a nice golden brown color on both sides should take about 10 minutes. Don't over pack the pot.
Remove the chicken and set it aside. Add the diced onion, carrot, celery and the remaining third of the salt. Fry until the onions get a glassy translucent look, about 3 -5 minutes.
Set the heat to medium-low then slowly caramelize the onions while stirring every now and again, for about 15 - 20 minutes.
Increase the heat then mix in the mushrooms. Fry until the mushrooms get browned a bit, for about 3 minutes.
Add dry white wine and water.
Put the pieces of chickens with the juice back, add the parsley, thyme and bay leaf. Add more water if the meat isn't covered, at least ¾ by the liquid. Stick the herbs under the liquid too. Bring it to boil.
Lower heat to simmer, place the lid on and cook until the thickest part of the chicken reaches the internal temperature of 165°F / 74°C that should happen in about 25 - 30 minutes.
Remove pot from heat then remove chicken from the pot. Discard the herbs.
Whisk egg yolk and cream in a bowl. While whisking constantly, spoon in some hot sauce from the pot. The sauce will heat up the liaison that is our thickening agent.
Stir the warmed up eggs and cream mixture in.
Mix in tarragon, lemon juice then put back the chicken pieces and the juice.