For the doughnuts, sift flour into a bowl. Dissolve yeast with some of the sugar in the lukewarm milk and add to the flour. Add remaining sugar, yolks, vanilla sugar, salt, lemon sugar, rum and butter also to flour and knead with a food processor or hand mixer until smooth.
Cover with plastic wrap and let rise to double height. Sprinkle flour on work surface, place dough on top, cut off pieces of equal size, drag round and let rise on floured baking sheet, covered, to double height.
Heat oil in a wide pan, bake doughnuts bottom side up for 2-3 minutes with lid closed, then turn and finish baking uncovered.
Pour jam into a piping bag with a long nozzle and fill the doughnuts with it. Sprinkle with powdered sugar.