Combine the mashed potatoes and 1/4 teaspoon salt in a large bowl. Add 4 cups confectioners' sugar and stir with a wooden spoon. Stir in the remaining 2 to 4 cups confectioners' sugar, 1 cup at a time, until the dough is stiff and dry. (The amount of sugar will depend on how starchy your mashed potatoes are.) Knead with your hands to bring the dough together.
Dust a large sheet of parchment paper with confectioners' sugar. Transfer the dough to the parchment and roll into a 12-by-14-inch rectangle, about 1/4 inch thick, dusting with confectioners' sugar if the dough gets sticky. (Refrigerate for a few minutes if the dough gets too soft to roll.) Spread the peanut butter evenly over the dough. Spread a long narrow strip of umeboshi paste on top of the peanut butter, about 3 inches in from one of the long ends. Refrigerate 30 minutes.
Starting with one of the long ends, roll up the dough like a jelly roll, using the parchment to help you. Turn seam-side down and slice into thin pieces.
Photograph by Kang Kim
Recipe courtesy Justin Warner for Food Network Magazine