In a bowl, combine flour, baking powder, baking soda and salt. Set aside.
In another bowl, beat eggs and sugar with an electric mixer for about 1 minute or until frothy. Add butter and mashed potatoes and mix thoroughly. Add dry ingredients and stir until dough is smooth. Form two discs of dough with your hands. Cover with plastic wrap and refrigerate for 1 hour.
Preheat shortening or oil in the deep fryer to 190 °C (375 °F). Line a baking sheet with paper towels. Place a wire rack on another baking sheet. Pour maple syrup in a bowl. Set aside.
On a floured work surface, roll out dough to an 8 to 10-mm (½-inch) thick sheet. Cut doughnuts with a 7,5-cm (3-inch) round cookie cutter and then cut out doughnut holes with a small 2-cm (¾-inch) round cutter. Reuse scraps. Let rest for 15 minutes at room temperature to allow dough to dry and form a nice thin crust while frying.
Place 3 or 4 doughnuts at a time into hot fat and fry for 2 to 3 minutes or until golden brown, turning halfway through cooking. Beware of splattering. Drain on paper towel and let cool for 1 minute. Soak doughnuts, one by one, in maple syrup. Drain on wire rack and let cool. Proceed the same way for doughnut holes (cooking only 1 minute).
Serve doughnuts warm or at room temperature, ideally the same day.
These doughnuts will keep for 2 days in an airtight container in a cool, dry place.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.