Begin by peeling and slicing the potato into thin pieces. Place the potato slices in a small saucepan and cover them with water. Cook over medium heat until the potato slices are soft and easily mashed. Once cooked, drain the potato slices and transfer them to a mixing bowl.

Using a potato masher or fork, mash the cooked potato slices until they form a smooth consistency. Ensure there are no lumps remaining in the mashed potato.

Add half of the confectioner's sugar to the mashed potato and mix well until fully incorporated. Then, add the softened salted butter and vanilla extract to the mixture. Blend everything together until smooth and evenly combined.

Line a clean countertop with waxed paper and lightly dust it with confectioner's sugar to prevent sticking. Transfer the potato mixture onto the waxed paper and sprinkle more confectioner's sugar on top. Use a rolling pin to roll out the dough to a thickness of about 1/4 inch.

Once the dough is rolled out, evenly spread a layer of creamy peanut butter over the surface, leaving a small border along the edges.

Starting from one end, carefully roll up the dough into a log, similar to rolling a cinnamon roll. Ensure the roll is tight and compact. Once rolled, wrap the log tightly in the waxed paper.

Place the wrapped dough log in a large zipper-seal bag and refrigerate until firm, typically for at least a couple of hours or overnight. Chilling the dough will make it easier to slice later on.

Once the dough is chilled and firm, remove it from the refrigerator and unwrap it from the waxed paper. Use a sharp knife to slice the dough log into individual rounds, about 1/4 inch thick. Serve the potato candy slices immediately, and store any leftovers in the refrigerator for later enjoyment.