Place potato into a pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until very tender, about 30 minutes. Drain. Transfer potato to a large bowl and mash until smooth. Stir vanilla extract into mashed potato.
Stir confectioners' sugar gradually into mashed potato until consistency is like cookie dough; add more confectioners' sugar if candy is too runny.
Sprinkle a large piece of waxed paper with confectioners' sugar and roll potato candy out into a rectangle 1/4 inch thick. Spread peanut butter onto candy. Roll candy and peanut butter into a tight spiral; wrap candy log in waxed paper. Refrigerate until firm, about 1 hour. Cut into slices to serve.