Drop the potatoes into a large pot of boiling water and cook until easily pierced with a fork, about 15 to 20 minutes. Cool slightly; then peel.
In a heavy 10 to 12-inch skillet, melt the sugar over low heat. Cook slowly until the sugar turns to a light brown caramel, 3 to 5 minutes. Stir constantly with a wooden spoon and watch the sugar closely; the syrup changes color very rapidly and burns easily. It must not become too dark or it will become bitter.
Stir the melted butter into the sugar and add as many potatoes as possible without crowding them. Shake the pan almost constantly to roll the potatoes and coat them on all sides with the caramel.
Remove the hot, caramelized potatoes to a heated serving bowl and repeat the procedure until all the potatoes are coated.