Melt 2 tablespoons of the butter in a large skillet. Add the chicken, season with salt and pepper and cook over moderate heat until slightly browned, about 4 minutes per side. Add the shallot and cook, stirring, for 1 minute. Add the Cognac and carefully ignite it with a long match. When the flames subside, add the Riesling, cover and simmer over low heat until the chicken breasts are just cooked, about 25 minutes. Transfer the breasts to a large plate and cover with foil. Cover and simmer the legs until cooked through, about 10 minutes longer. Transfer to the plate.
Meanwhile, in a medium skillet, melt 2 tablespoons of the butter. Add the mushrooms, season with salt and pepper and cook over low heat until the liquid evaporates, about 7 minutes. Increase the heat to moderate and cook, stirring, until browned, about 3 minutes.
In a bowl, blend the flour and the remaining 1 tablespoon of butter. Stir the cream into the large skillet; bring to a simmer. Gradually whisk the flour paste into the cooking liquid and simmer, whisking, until no floury taste remains, 3 minutes. Season with salt and pepper. Return the chicken to the skillet, add the mushrooms and briefly reheat.
Noodles, spaetzle, rice or boiled potatoes.
A complex Riesling with some body will blend with the cream and echo the wine here. Try one from Alsace or New Zealand.