Sprinkle the chicken pieces with salt and pepper.

Heat 1 Tablespoons butter and 1 Tablespoon olive oil in a skillet and brown the chicken over medium high heat, starting with the skin side down. This should take about 5 minutes on each side. Remove the chicken pieces.

Add the carrot, celery and onion to the pot and cook about 10 minutes, until softened.

Stir in the tablespoon of flour and continue stirring a few minutes, until the flour has been absorbed.

Stir in the white wine and cook a few minutes, until thickened.

Stir in the chicken stock and add the chicken pieces back to the skillet.

Add a sprig or two of fresh parsley and the bay leaf. Bring to a simmer, cover partially and cook the chicken for about 30 minutes, until cooked through.

While the chicken is cooking, use a separate skillet to saute the mushrooms in a tablespoon of butter and set aside. Salt lightly.

When the chicken is done, remove it to a platter and take the parsley and bay leaf out of the sauce. Continue simmering the sauce a few more minutes, until thickened.

In a small bowl, whisk together the egg yolk and cream. Add a few spoonfuls of the hot sauce and stir to temper it. Pour this mixture back into the sauce, stirring gently and let it cook a minute or so.

Add the chicken, skin side up, and cooked mushrooms back in, then stir in a tablespoon or two of fresh lemon juice and another tablespoon of butter.