Remove the excess moisture from the chicken with paper towels. Sear the chicken in peanut oil until healthily brown. Degrease the pan. Melt more butter in the pan and sprinkle flour in gradually - while stirring with a whisk. Whisk constantly until the butter/flour mixture becomes golden brown, and emits a nutty aroma. Introduce carrot, onion, garlic, peppercorns and bay leaf into the pan. Reduce the flame to medium-low and saute vegetables until soft - approximately 6-7 minutes. Pour wine into the pan, gradually in stages, while whisking constantly. Add the thyme. Return the chicken to the pan. Season with salt. Bring the contents of the pan slowly to a boil. Seal the pan with a tightly fitting lid and braise in the oven for 35-40 minutes.
Remove chicken from the pan. Strain the remaining contents of the pan through a fine mesh collander or chinois, pressing very lightly on the debris with a wooden spoon. Discard the debris. Return both the chicken and the velvety sauce to the Dutch oven. Simmer covered over very low heat.
Peel the pearl onions. Quarter the mushrooms. Carve the fatback into 1/2 inch wide lardons. About 10 minutes before straining the sauce, blanch the fatback in boiling water for 5 minutes. Strain the fatback out of the water, and throw it into a medium hot skillet. Carmelize every side of every lardon. Remove the fatback. Add the pearl onions. When golden brown, add the mushrooms. Season with salt and pepper. Do not overcook mushrooms. (If the fatback refuses to render a sufficient quantity of fat for sauting onions and mushrooms, add a knob of butter.) Throw the contents of the skillet into the Coq au Vin. Deglaze the skillet with wine, reduce and add it to the Coq au Vin. Heat the contents for approximately 4 minutes.
Arrange the chicken on a platter, drench it with sauce and scatter the garnish over of the chicken. Serve with parsley new potatoes, rice pilaf or buttered noodles.