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Drain and cool the potatoes to room temperature.
Preheat the broiler.
Peel and cube the potatoes, then place in a large pot set over medium-low heat. Add the butter, milk, shredded cheese and some salt and pepper. Cook, stirring gently until the cheese is melted and hot, about 45 minutes.
Portion the potatoes onto individual serving dishes. Cover each serving dish with some sliced Cheddar (about 1 1/2 slices per serving) and place under the broiler until melted and slightly brown. Serve hot.