For the potato-hazelnut doughnuts, cut the cold butter into small cubes and crumble with flour and salt. Press the cooked, cold potatoes through a potato press or grate them on a grater and knead them with the butter to a dough.
Roll out the dough into a square and fold it into three parts (like a puff pastry). Roll out, fold in three again and repeat. Then roll out the dough to a thickness of about 1 cm and cut out doughnuts with a round cutter.
Beat the egg with a little water and brush the doughnuts with it. Sprinkle with the coarsely chopped hazelnuts and place the potato-hazelnut doughnuts on a baking tray covered with baking paper. Bake in the preheated oven at approx. 200°C until golden brown.