Cook shallot in 1 tablespoon butter in a small heavy saucepan over moderate heat, stirring occasionally, until softened, about 2 minutes. Add wine and boil until most of liquid is evaporated, about 3 minutes. Set aside.
Pat chicken dry and sprinkle all over with 1/2 teaspoon salt and 1/4 teaspoon pepper. Heat oil in a 12-inch heavy skillet over moderately high heat until hot but not smoking, then sauté chicken in 3 or 4 batches, turning over once, until golden and just cooked through, about 2 minutes per batch. Transfer as cooked to a platter and keep warm, loosely covered with foil.
Return shallot mixture to moderately low heat and add remaining 3 tablespoons butter, 1 cube at a time, whisking until incorporated. Remove from heat and whisk in tarragon, parsley, lemon juice, and remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. Spoon sauce over chicken.
