For each fine chicken:

If using a whole chicken, cut the backbone and neck out and then cut off the wing tips. Use them for stock. Cut the rest of the chicken into two drumsticks, two thighs. Cut the whole breast into 3 or 4 pieces. Leave the bones in, but you can trim off excess skin or fat globs.
To make the stock, put the chicken backbone, neck, wingtips, giblets and any trimmings of bone, skin or fat into a deep pan, cover with water, add the quartered unpeeled onion and a carrot, half a dozen whole peppercorns and bring to the boil. Turn the heat down and let it simmer uncovered until reduced to about 2 cups.
Cut the bacon/pancetta into short strips; they need to be about the diameter of a BIC pen. Simmer for 5 minutes and drain to remove some of the salt. Put them, together with the butter, into a thick-bottomed casserole - if you are going to finish in the oven, one of enamelled cast iron would be perfect - and let them cook over a moderate heat. Stir the pancetta - it mustn't burn - then, when it is golden, lift it out into a bowl, leaving behind the fat in the pan.
If you are not using bacon, cook 3 tablespoons of butter in the casserole over a medium heat until it just begins to brown.
Season the chicken pieces with salt and pepper and place them in the hot fat in the casserole, so that they fit snugly, but have room to color. Turn them when the underside is pale gold, then cook the second side to the same degree. This will take about 12-16 minutes. The skin should be a golden or honey color, not brown. Lift the chicken out and into the bowl with the pancetta. The thin layer of browning on the pan holds much of the flavor of the dish, you should have a thin film of goo starting to stick to the pan.
Now add the large cut up onion, optional carrots and celery, and cook about 10 minutes. Cook slowly, stirring from time to time, until the onion is translucent and has released enough moisture to dissolve some of the pan stickings. Add the optional garlic, peeled and thinly sliced, and cook no more than 5 minutes.
If you are making the dish to eat the same day, melt 2 tablespoons of butter in a separate pan. Add pearl onions and cook for 5-8 minutes until lightly browned. Add mushrooms, halving or quartering them if they are too big. Season with salt, cover and cook for 5 minutes until mushrooms release their liquid. Remove cover, increase heat to high and cook, stirring, until liquid has evaporated and onion and mushrooms are golden brown, about another 5 minutes.
NOTE: If you are making the chicken ahead, the mushrooms and pearl onions are added the next day when the chicken is reheated!
Return the chicken and pancetta to the pan, stir in the optional flour and optional tomato paste, add the onions and mushrooms and let everything cook for a minute or two before pouring in the cognac, wine and tucking in the herbs.
Spoon in ladles of the simmering chicken stock until the entire chicken is covered. Bring to the boil, then, just as it gets there, turn the heat down so that the sauce bubbles gently. Cover loosely with a lid.

To cook on the stovetop: simmer gently, partially covered. Turn chicken once or twice as necessary during cooking, until tender and infused with wine flavor. Check after 40 minutes. It could take up to 60 minutes.

To cook in the oven or reheat: Put the oven to 325. Put the covered chicken in for 50-75 minutes, stirring once or twice, until tender but not shredded.

When the chicken is cooked, transfer chicken and vegetables to a platter and cover with aluminum foil to keep warm. Strain sauce through fine wire mesh sieve set over large bowl, pressing on solids with a spoon to release all the liquid. Sauce should measure 2-3 cups. Return sauce to pan.
To thicken by reduction, uncover and turn up the heat. As it bubbles down it will become thicker - though not thick - and will become glossy. Cook uncovered until the sauce is reduced to about 2/3 the original volume.
To thicken with buerre manié (butter and flour): counting 1 tablespoon each of butter and flour for each cup of sauce, mash 2 to 3 tablespoons each butter and flour in small bowl to make a beurre manié. Bring sauce to boil and whisk in beurre manié until thickened and smooth. Cook at least 5 minutes, stirring attentively, to reduce the raw flour taste.
An authentic variation is to crush the cooked liver in a vegetable mill or liquidise lightly, then add to sauce, mixing thoroughly.

To serve: Add reserved chicken, bacon, onions and mushrooms back to the thickened sauce; adjust seasoning with salt and pepper to taste. Reduce heat to medium-low and simmer very gently to warm through and blend flavors, about 5 minutes. Check seasoning again and adjust if necessary; add parsley. Serve sauce and chicken from the casserole or transfer chicken to serving plate; pour sauce over chicken. Serve immediately.

Quantity preparation note:
if you are making a lot of sauce, 3 or more chickens, you can PRE-Concentrate the sauce to avoid the straining. Bring all the red wine and prepared chicken stock to boil in a large, heavy saucepan. Reduce heat to medium-high and simmer until reduced, about 20 minutes. 1 bottle (3 cups or 750ml) fruity red wine PLUS 2 1/2 cups chicken stock should cook down to about 4 cups; this is enough for 1 big chicken.