Peel the potato, cut it into small chunks, and place in a saucepan with enough water to cover. Boil until tender and easily pierced with a fork. Drain, mash thoroughly, and measure out ½ cup of the mashed potato. Transfer it to a large mixing bowl and let it cool completely to room temperature.
Gradually mix in the powdered sugar, adding one cup at a time. After about six cups, check the consistency—if the dough holds together in a firm ball, stop adding sugar. If it’s still sticky, continue adding more until the dough is firm and can be rolled out easily.
Shape the dough into a ball, then wrap it tightly and refrigerate for 40 to 60 minutes. Be careful not to chill it too long, as this could make it brittle.
Dust a flat surface liberally with powdered sugar. Divide the chilled dough in half and roll one portion into a rectangle about ⅓ inch thick. Trim any uneven edges.
Spread an even layer of peanut butter over the dough, making sure to leave about ½ inch of space around the edges.
Carefully roll the dough into a tight log, starting from one of the longer sides.
Repeat the process with the second half of the dough.
Using a sharp knife, slice the logs into ½-inch thick pieces. If the dough feels too soft while cutting, refrigerate the logs for a short time before slicing. Store the finished candy in an airtight container in the refrigerator.