Put the chicken in a bowl with the garlic, one leaf/sprig of each of the herbs and a scant teaspoon of coarse salt. Pour in the wine and leave to marinade for 12 hours.
Meanwhile, put the carcass in a deep pan with the rest of the herbs, the carrot and onion. Half-cover with cold water and bring to the boil, then turn down the heat, skim and cook very gently for an hour and a half. Strain, discarding the bones and aromatics (after picking any meat off), and reduce the stock to about 300ml.
The next day, remove the chicken from the wine and pat dry, then set aside. Melt a quarter of the butter in a large casserole pan over a medium-high heat and fry the lardons until golden. Scoop out with a slotted spoon, add another quarter of the butter and fry the leeks with a pinch of salt until soft. Scoop out with a slotted spoon and add to the bacon. Repeat with the mushrooms.
Melt the remaining butter and season the chicken. Fry until deep golden on all sides, then scoop out, turn down the heat and add the flour. Cook for a couple of minutes then stir in the wine, scraping the base of the pan. Add the stock and the chicken, bring to a simmer, then cover, turn down the heat and cook for an hour and a half, until the meat is falling from the bones. Take the chicken out of the pan.
Whisk together the cream and egg yolk, then take the pan off the heat and whisk this into the sauce. Put it back on a gentle heat and bring slowly to a simmer, then cook until it lightly coats the back of a spoon, stirring regularly. Season to taste, then stir in the lardons, leeks, mushrooms and chicken.