Use completely cooled, plain mashed potatoes. If making them fresh, boil and mash a peeled potato until smooth, then let cool. Avoid adding any butter, salt, or milk.
In a large bowl, combine the mashed potatoes and vanilla extract. Gradually mix in powdered sugar, one cup at a time, until a thick, pliable dough forms. It should be soft but not sticky. You may need up to 6 cups depending on the moisture in your potatoes.
Place a sheet of wax or parchment paper on a clean surface. Dust with powdered sugar and roll the dough into a rectangle about ¼ inch thick.
Evenly spread the peanut butter across the dough, leaving about ½ inch border around the edges.
Carefully roll the dough into a tight log, starting from the long edge. Wrap the log in parchment or plastic wrap and chill for at least 1 hour until firm.
Once chilled, unwrap the log and slice into ½-inch rounds. Serve immediately or store for later.