Rub the chicken with the cut side of one-half lemon.
Lay the bacon slices across the breast to cover, then truss the chicken closed with twine and reserve.
Scatter 2 tablespoons butter and the diced bacon across the bottom of a flameproof large casserole with cover.
Add the carrots, onion, celery, parsley and chicken giblets.
Cover and cook over very gentle heat to sweat until tender without letting the vegetables color, a scant 30 minutes (If necessary, add 1 tablespoon water from time to time).
Once the vegetables have softened enough, add just enough water to cover them.
Lay the chicken breast side up on top of the vegetables. Add the stock and enough of the wine to cover the bird by about half an inch and bring to a simmer over moderate heat.
Use a spoon to skim the little bit of foam that rises to the surface when the liquid starts to boil.
Add the cut stems and any trimmings from the mushrooms, the clove and a scant 2 teaspoons salt.
Cover the casserole, leaving the lid slightly ajar for steam to escape.
From this point, the liquid should not boil but barely simmer, so that the chicken is cooked by poaching for 45 to 55 minutes. Check that the chicken is done by piercing the flesh of the drumstick joint, which takes the longest to cook--the pearl of juice released must be completely white.
While the chicken is cooking, prepare the mushroom garnish.
Place the mushroom caps in a small saucepan with one-half cup water, a pinch of salt, the juice of the remaining half lemon and 2 tablespoons butter.
Cover, bring to a boil and cook 4 to 5 minutes, shaking the pot from time to time. Remove from the heat, but leave covered to keep warm and drain off the water before serving.
To finish the dish:
Strain the chicken cooking liquid into a bowl, discarding the vegetables.
Leave the fowl in the covered pot off heat to keep it good and hot while preparing the sauce.
Measure 3 cups of the cooking liquid and boil it rapidly in a sauté pan to reduce it by at least half, to about 1 1/4 cups (You will have extra stock which you can freeze and use for other things).
Cool the reduced stock in a bowl.
In the same pan, melt the remaining 2 tablespoons butter over low heat, stir in the flour to make a smooth paste and cook for 2 minutes.
Remove the pan from the heat and let the roux cool for a minute or two.
Whisk in the reduced cooking liquid a little bit at a time, whisking continuously. When all the broth has been added, return the pan to the fire over high heat and cook for another couple of minutes.
Stirring continuously, add the cream gradually, in 4 or 5 additions, adding the fresh amount only when the sauce has resumed boiling; this takes just a few minutes (the sauce should be perfectly smooth and just thick enough, with a beautiful white matte tint, to justify its name).
The sauce can also be kept warm in a double boiler until you are ready to serve.
To serve: You should always carve this dish in the kitchen because the skin should be removed from the pieces.
Take off the bacon strips.
Detach the thighs and separate the drumstick from the upper thighs.
Detach the wings at the same time that you cut the breast.
Remove the skin from the legs and breast and discard.
Divide each wing into 2 pieces cut on the bias--you will have 6 pieces in total.
Arrange the chicken on a round plate that has been well warmed and arrange the mushrooms around the outside.
Cover each piece of chicken with a few tablespoons of sauce and serve the rest in a sauceboat.