Combine confectioners' sugar, mashed potatoes, butter, and vanilla extract in a large bowl until a thick dough forms. Refrigerate dough until chilled, about 30 minutes. Stir extra confectioners' sugar into dough if too moist.
Spread a large sheet of waxed paper on the counter. Sprinkle with confectioners' sugar.
Dust rolling pin with confectioners' sugar. Roll chilled dough on prepared waxed paper into a 1/4-inch thick rectangle.
Spoon peanut butter into a microwave-safe bowl. Heat in microwave on High until slightly softened, 20 to 30 seconds.
Spread softened peanut butter onto dough.
Roll dough, starting from long end, into a jelly roll shape.
Wrap the roll in waxed paper and refrigerate until firm, about 2 hours. Slice into 1/4-inch slices.