Using a mandoline, cut the potatoes and apples into 1/8-inch slices. Stack the slices and cut them into 1/8-inch sticks, then cut the sticks into 1/8-inch cubes.
In a large skillet, melt 1 tablespoon of the butter. Add the garlic and onion and cook over moderate heat, stirring, until softened, about 4 minutes. Add the wine and simmer over moderate heat until almost evaporated, about 4 minutes. Add the potatoes, apples and chicken broth and bring to a boil over moderately high heat. Reduce the heat to moderate and simmer, stirring occasionally, until the potatoes are tender and the liquid has reduced to 1/4 cup, about 25 minutes. Remove from the heat. Season with salt and pepper and stir in the remaining 2 tablespoons of butter until melted. Stir in the Parmesan and serve right away.
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