Season chicken pieces on both sides with salt and pepper. Heat a few tablespoons of clarified butter in a Dutch oven over high heat. Place chicken in skin side down and cook until golden brown. Turn the chicken over and brown the other side. Remove chicken to a baking sheet and bake in the oven until just cooked through, about 12 to 15 minutes.
Remove all but 2 tablespoons of the fat from the pan. Add the mushrooms, season with salt and pepper, and saute until golden brown. Add the shallots and cook for 30 seconds. Remove the pan from the heat and add the Cognac and cook until completely reduced. Add the wine and cook until completely reduced. Add the stock and tomato and bring to a simmer. Cook until the sauce is thick enough to coat the back of a spoon ("nappe") and then swirl in the butter ("monter au beurre"). Once the butter is added, the sauce can not be further reduced because it will break. Season with salt and pepper and stir in the tarragon and chervil.
Remove the chicken from the oven and let rest for 5 minutes. Remove the breastbone from the breast. Cut each breast into 2 even pieces on the bias. Cut the legs into 2 pieces at the joint and on the bias and remove the thighbones. Serve 1 leg piece and 1 breast piece. Make sure only 1 piece contains a bone. Serve on hot plates, napped with the sauce.