In a large heavy skillet, melt the butter in the olive oil. Season the chicken pieces with salt and pepper. Add them to the skillet skin side down and cook over moderately high heat, turning once, until browned, about 12 minutes; transfer to a plate.
Discard all but 2 tablespoons of the fat from the skillet. Add the mushrooms and cook, stirring occasionally, until they release their liquid, about 5 minutes. Raise the heat to high and cook, stirring occasionally, until the mushrooms are golden, about 5 minutes longer; transfer to a bowl.
Add the shallot to the skillet and cook over moderately high heat until barely soft, about 30 seconds. Return the thighs and drumsticks to the skillet. Add the cream, sherry and tarragon and bring to a boil. Cover and simmer over low heat for 20 minutes. Add the breasts, wings and mushrooms, cover and simmer for about 15 minutes longer. Transfer the chicken and mushrooms to a platter, cover and keep warm.
Increase the heat to high and boil the sauce, stirring, until it is thickened, about 5 minutes. Remove from the heat and whisk in the truffle oil. Season with salt and pepper. Pour the sauce over the chicken and serve.
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