In a bowl, combine the milk and vinegar. Set aside.
In another bowl, combine the flour, baking powder, baking soda, and salt. Set aside.
In a third bowl, cream the butter with the sugar and mashed potatoes. Add the eggs, one at a time, and beat until smooth. At low speed, add the milk mixture alternately with the dry ingredients. The dough will be very sticky. Cover with plastic wrap and refrigerate for at least 1 hour or overnight.
Preheat the oil in the deep fryer to 190 °C (375 °F). Line a baking sheet with a few layers of paper towels.
On a generously floured work surface, spread half of the dough at a time into a 1-cm (½-inch) thick sheet, with your hands. With a 4-cm (1 ½-inch) round cookie cutter or a small shot glass, cut discs of dough.
Fry the doughnut holes, 6 to 8 at a time in the oil for about 1 minute on each side or until golden brown. Drain on the baking sheet. If desired, roll in sugar, to coat. Let cool. Store in an airtight container for about 2 days or freeze and thaw to room temperature.