How to Make It
Season the chicken breasts with salt and pepper. In a large skillet, melt 1 tablespoon of the butter in the vegetable oil. Add the chicken breasts, skin side down, and cook over moderately high heat until lightly browned, about 4 minutes. Cover the skillet and cook the chicken over moderate heat until the skin is crisp, about 4 minutes. Uncover, turn the chicken and cook over moderately low heat until the meat is just white throughout, about 6 minutes. Transfer the chicken to a large plate and cover loosely with foil.
Pour off the fat from the skillet and add the remaining 1 tablespoon of butter. Add the mushrooms and shallot and cook over moderate heat, stirring, until softened, about 4 minutes. Add the wine and Cognac and simmer over moderately high heat until almost dry, about 3 minutes. Add the cream and stock and simmer until thickened, about 8 minutes. Stir in the mustard and dill and season with salt and pepper. Add the chicken to the sauce, skin side up, along with any accumulated juices. Simmer over low heat until the chicken is warmed through, about 2 minutes. Spoon the sauce onto plates, top with the chicken and serve.