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Preheat oven to 400 degrees.
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Place 1 tbsp butter in a .5 quart oven-safe dish. Place in oven while it's preheating, and allow butter to melt. Meanwhile, peel and slice potatoes into thin circles.
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Once butter is melted, coat the dish in butter, and place a layer of thinly-sliced potatoes on the bottom of the dish. Sprinkle on 1/3 of the Parmesan cheese, and repeat three layers.
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In a small saucepan, melt 2 tbsp butter over medium-heat. Sprinkle in flour, and whisk quickly to avoid any lumps. Add in milk, whisking vigorously, until it starts to thicken. Stir in cheese and remove from heat.
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Pour cheese sauce over potato dish. Bake on the bottom rack in preheated oven for 60 - 80 minutes, until potatoes are tender and cheese is golden brown.