Grate the potatoes with a normal cheese grater. Let rest so they release some of their water. Squeeze to remove even more water, then add the speck (cut into thin strips), salt, pepper and the egg. Mix well. Use your hands to create discs that are 1 cm thick and 8-10 cm in diameter. In a saucepan, melt the butter. Once nice and hot, place the potato rösti in the pan. Cook until golden brown all over. Try not to move or stir the rösti too much so they won’t break. Separately, chop or tear the mint and add it to the strained yoghurt to create a rich, refreshing topping. Serve the rösti warm, accompanied by thin speck slices and the yoghurt sauce.