For onions
For mushrooms
For chicken
To finish
Peel onions: Trim the end off of the basal plate of each onion (don't cut it completely off), and then peel outer dry layers. Put peeled onions, butter, chicken stock, seasonings and herb bouquet into a medium saucepan. Cover and bring to a simmer over low heat. Cook very gently for 45 to 50 minutes, adding liquid as needed, but don't stir. Set aside.
Bring the stock, butter, lemon juice, and salt to boil in a small saucepan. Add the mushrooms and stir. Cover and cook over medium high heat for 5 minutes. Set aside.
In a large enameled cast iron pot, cook the vegetables very slowly in the 4 tablespoons of butter until almost tender but not browned.
Meanwhile, cut chicken into pieces (thighs, drumsticks, wings, breasts), leaving bones in and skins on (reserve back for making stock). Cut breasts in half crosswise.
A few at a time, pat the chicken pieces dry and add to pan, nestling into softened vegetables. Sauté, turning every minute, for 3-4 minutes. Do not brown; you don’t want to completely cook the chicken yet (the point is a slow cook for a delicate sauce with a balanced flavor). When the meat has stiffened a bit and the skin is yellow, remove the chicken pieces and set aside in a large bowl.
When all chicken has been sautéed, put all the chicken pieces back into the pot with accumulated juice. Cover and cook very slowly for 10 minutes, turning the pieces once. The chicken meat should stiffen more and swell lightly, but not gain any more color. Sprinkle salt, pepper, and flour over both sides of the chicken pieces. Roll each piece to coat the flour with the melted butter at the bottom of the pan. Cover and cook slowly for 4 minutes, turning once.
Pour 3 cups of hot stock into the pot and shake to mix the flour into the liquid. Add wine and herb bouquet. At this point, the liquid should almost cover the chicken pieces (add more hot stock if it doesn't). Bring to a simmer and taste/adjust liquid for salt. Cover and cook at a low simmer for 40 minutes (or longer, depending on the chicken). Chicken is done when meat is tender and juices run clear yellow. When cooked, remove the chicken pieces to a bowl and reserve. I like to take the breasts out a little early as they can dry out before the other pieces cook.
Uncover the pot and increase heat to medium. After a few minutes, skim fat, then crank it up. Boil liquid until it is reduced to about 3 cups. Turn heat to low.
Meanwhile whisk the egg yolks and heavy cream in a medium heat-proof bowl. A few spoonfuls at a time, whisk some of the hot sauce into the egg mixture, until about a cup has been added. Whisk egg mixture in thin stream back into the pot. Simmer, whisking constantly, for one minute, then remove from heat. Strain sauce through fine-mesh strainer. Adjust seasoning if necessary. Note: Julia's recipe recommends adding a teaspoon of lemon juice to the sauce here. I like the flavor, but even a few drops of lemon invariably makes the sauce curdle, so I omit it.
At this point, everything can be held at room temperature for a couple of hours.
About 30 minutes before serving, preheat oven to 300°F. Arrange chicken pieces, mushrooms, and onions in (cleaned) pot, and pour sauce over. On stovetop, bring to a simmer. Cover and put the pot into the oven. Bake until chicken is warmed through.
To finish:
Remove pan from oven, and drop 2 Tablespoons butter onto the sauce. Spoon the sauce over the butter until it melts and has become incorporated. Serve with buttered egg noodles or rice. Garnish with parsley sprigs (optional).
