3 pounds baking potatoes, peeled and thinly sliced on a mandoline
1/2 cup freshly grated Parmesan cheese (about 2 ounces)
In a large, deep skillet, melt the butter in the oil. Add the brussels sprouts and shallot and cook over moderate heat, stirring, until softened, about 8 minutes. Add the white wine and cook until evaporated, 1 minute. Add the heavy cream, season with salt and pepper and bring to a simmer. Let cool to room temperature.
Preheat the oven to 325°. Butter a 1 1/2-quart shallow baking dish. Arrange one-sixth of the potato slices evenly over the bottom of the dish and season with salt and pepper. Using a slotted spoon, spread a thin layer of the brussels sprouts over the sliced potatoes and top with 1 tablespoon of the Parmesan cheese. Repeat the layering process, using all of the potatoes and brussels sprouts and all but 2 tablespoons of the Parmesan and ending with a layer of potatoes. Pour the cream over the potatoes and sprinkle with the remaining 2 tablespoons of Parmesan. Cover with buttered foil and bake for 45 minutes.
Uncover and bake the gratin for 30 minutes longer, or until the potatoes are tender and the top is golden brown. Let sit for 15 minutes before serving.