A black iron skillet is essential for the authentic preparation of this dish. Sprinkle the chicken on both sides with salt and pepper. Select a skillet large enough to hold the chicken comfortably when it is opened up, as for broiling. Fold the chicken wings under to hold them secure.
Melt 2tablespoons butter in the skillet and add the chicken, skin side down. Cover the chicken firmly with a plate that will fit comfortably inside the skillet. Add several weights, approximately 5 pounds, to the top of the plate. Cook over low heat until the skin side of the chicken is nicely browned, about 25 minutes.
Pour off most of the fat from the skillet and add the remaining 2tablespoons butter. Turn the chicken skin side up. Scatter the mushrooms over the chicken. Scatter the onions around it. Add the bay leaf, thyme, wine, and chicken stock. Replace the plate and the weights and continue cooking over low heat about 45 minutes longer.
Remove the chicken to a warm platter. Spoon the mushrooms and onions around the chicken. Cook down the skillet liquid until reduced by half. Pour this over the chicken and sprinkle with parsley.