Melt the butter in a Dutch Oven. Gradually, sprinkle the flour into the grease. Stir constantly with a whisk until the roux is blonde in color and offers up a slightly nutty aroma. Introduce bay leaf, carrot, celery, onion, garlic and peppercorns into the pan, and saute the vegetables until soft, about 8 minutes. Pour in the stock. Whisk until roux is perfectly incorporated into the liquid. Heat up the skillet used to brown the chicken. De-glaze it with wine. Scrape the bottom of the skillet to dislodge the residue. Pour the contents of the skillet into the Dutch oven. Place the chicken into the pan. Sprinkle in the thyme and cayenne pepper. Season with salt. Bring the contents of the pan slowly to a pulsating boil. Seal the pan with a tightly fitting lid and simmer until chicken is very tender, about 1 hour.
Skim the surface of the liquid to remove any scum or grease arising on the surface. Remove the chicken from the pan. Lightly strain the remaining contents in the pan through a fine mesh collander or chinois. Return the sauce to the Dutch oven. (If using mushrooms, saute them in butter, add them to the pan, and simmer until well warmed through.) Check for consistency. If the sauce does not coat the back of a spoon, reduce it. Return the chicken to the pan and warm the chicken pieces through. Cut off the flame and allow the chicken to rest for 10 minutes, covered.
Below is an alternate recipe (less meat plus shallots):
Melt the butter in a Dutch Oven. Gradually, sprinkle the flour into the butter. Stir constantly with a whisk until the roux is blonde in color and offers up a slightly nutty aroma. Add the shallots to the pan and sweat them down over a medium-low flame. Throw in the mushrooms and saute them until all of their water evaporates. Add the garlic and saute for 30 seconds. Pour in the wine and stock, slowly. Whisk until the roux is perfectly incorporated with the liquid. Return the chicken to the pan. Season the gravy with salt, pepper, ground cayenne pepper, and thyme leaves. Bring the liquids to a boil, cover the pan, and cut the heat top a slow simmer. Braise for 40-45 minutes.
If, after 40-45 minutes, the sauce does not coat the back of a spoon, remove the chicken pieces and reduce it. Return the chicken to the pan and warm the chicken pieces through. Cut off the flame, allowing the chicken to rest for 10 minutes, covered. Serve with rice.