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In a medium mixing bowl, combine cold mashed potatoes with salt and vanilla.
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Add icing sugar, one cup at a time, beating after each addition.
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At first the mixture will be stiff and hard to handle but with beating it will become very runny.
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With the addition of the fourth cup, the mixture will firm up again.
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Turn half of the mixture on to a board that's been lightly dusted with icing; roll into a rectangle two inches thick.
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Spread with half of the peanut butter.
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Roll up from the short side, jelly-roll fashion.
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Repeat with rest of fondant.
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Refrigerate until firm; slice 1/4 inch thick.