Melt 2 tablespoons of butter in a large skillet over medium-high heat. Add chicken and season with salt and pepper and cook until somewhat golden browned, about 4 minutes per side. Add shallot and continue cooking and stirring for 1 minute. Add Cognac and carefully ignite it with a long match. When the flames subside, add Riesling, cover and simmer on low until the chicken breasts are just cooked, about 25 minutes. Remove breasts to plate and cover with foil. Cover and simmer the legs until cooked through, about 10 minutes longer. Transfer to the plate.
Meanwhile, in another skillet, melt 2 tablespoons of the butter. Add mushrooms, season with salt and pepper and cook over low heat until the liquid dissipates, about 7 minutes. Increase heat and continue cooking and stirring, until browned, about 3 minutes.
In a bowl, blend flour and remaining tablespoon of butter. Stir 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.