Warm the milk (approx. 30°).
Dissolve yeast in 2 tablespoons of the milk, stir with a little of the flour to form a thick paste. Sprinkle this pre-dough with flour, cover with a cloth and let rise in a warm place (ideal is 28 to 30°) until the volume has increased significantly and the surface shows cracks. (takes about 15 minutes) Mix the egg, egg yolk, salt, sugar, vanilla sugar, grated lemon zest, remaining milk and rum and whisk with the butter. Beat with remaining flour and pre-dough until smooth (food processor with dough hook).
Cover dough with a kitchen towel and let rest at room temperature for about 1 hour, it should rise well.
Shape doughnuts, place on floured kitchen towel and let rise again for 20 minutes.
Heat the shortening to 160° and bake the doughnuts on both sides.
