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1
Place potatoes in large pot with chicken broth and add enough water to cover potatoes completely; bring to a boil and cook until potatoes are tender but still firm, about 15 minutes. Drain potatoes, transfer to a bowl and refrigerate until chilled, 8 hours to overnight.
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2
Preheat oven to 400 degrees F (200 degrees C).
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3
Grate potatoes, discarding any lumps or skin that don't pass through the grater.
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4
Heat olive oil and butter in a large, oven-proof skillet over medium-high heat. Saute onion and mushrooms in hot oil mixture until soft but not browned, 5 to 7 minutes. Stir potatoes into skillet. Season potato mixture with salt and black pepper; stir gently and shape into a large potato cake in the skillet.
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5
Cook the potato cake in the skillet until base of cake is browned, 5 to 10 minutes. Place a plate on top of the skillet, turn the skillet upside-down to remove the cake from the pan, and slide cake, browned-side up, back into skillet. Cook until the other side of the cake is browned, 5 to 10 minutes more. Sprinkle Swiss cheese over the top of the cake.
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6
Bake in the preheated oven until cheese melts and starts to brown, 10 to 20 minutes. Cool in the skillet for 10 minutes before slicing.
