Preheat the oven to 425 degrees.
Pat the chicken breasts dry with paper towels and sprinkle them generously on both sides with salt and pepper. In a 12-inch cast-iron skillet, heat the oil over medium-high heat for 2 minutes, until it begins to smoke. Place the chicken breasts, skin side down, in the skillet and cook for 4 to 5 minutes without moving, until golden brown.
Using tongs, turn the chicken breasts skin side up, place the skillet in the oven, and roast for 12 to 15 minutes, until the chicken is cooked through.
Meanwhile, in a medium saute pan, combine the white wine, lemon juice, and shallots and cook over medium-high heat for about 5 minutes, until only 2 tablespoons of liquid remain in the pan. If it reduces too much, add an extra splash of white wine or water. Add the cream, 1 teaspoon salt, and 1/4 teaspoon pepper and bring to a full boil. Remove from the heat, add the diced butter, and swirl the pan until the butter is incorporated. Don't reheat or the sauce will "break"! Sprinkle with salt and serve the chicken hot with the sauce spooned over it.
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