Potato: Place cubed potatoes in a small saucepan and cover with water. Add 1/4 teaspoon salt. Bring to a boil and cook for 8 minutes or until fork tender.
Drain potato cubes and mash with 1 tablespoon butter until completely smooth. Place into the fridge and cool completely. (Mashed potato should equal 1/2 cup.)
In a large bowl, use a hand mixer or a stand mixer to whip the cold mashed potatoes with 2 Tbsp milk and vanilla extract. The mixture will look runny at first.
Add powdered sugar 1 cup at a time beating on medium speed repeating the process of adding one cup of sugar at a time, until a pliable dough forms. If it appeals crumbly, test the consistency by squeezing a small amount in your hand. It should hold together when squeezed and not be sticky.
Once it's ready use your hands to bring the dough together forming it into one large ball. Cover the bowl with plastic wrap and chill for 30 minutes to one hour.
Dust a large sheet of wax paper with powdered sugar. Dump 1/2 of the dough into the middle.
Roll into roughly a 8-inch or 9-inch square around 1/8-inch thick. You can use a sharp knife to trim the edges. Dust the top with powdered sugar and cover with a piece of wax paper. Flip the potato candy dough. Carefully peel the wax paper away. This will dust both sides making it easier to roll. Continue to dust with more powdered sugar while rolling, if it gets sticky.
Spread a thin layer of peanut butter from end to end using enough to cover leaving a 1/4-inch border. Begin to roll away from you using the wax paper ending seam side down. Wrap in wax paper and freeze for at least 1 hour or chill in the refrigerator overnight. (Repeat same process with the other half.)
Use a sharp knife to trim the ends, then slice into 1/2-inch slices. Chill until serving.