Put the flour, sugar, vanilla sugar, salt, egg and butter in flakes in a bowl and knead with the dough hooks of the hand mixer. Then knead with your hands to a smooth dough. Wrap in foil and chill for about 1 hour.
Roll out the dough on a floured work surface to a thickness of about 2 mm. Cut out 76 circles (5 cm Ø each). Place on 3 baking trays lined with baking paper. Bake one after the other in a preheated oven (electric range: 200 °C/ convection oven: 175 °C/ gas: level 3) for 7-9 minutes.
Remove, place on a cake rack and let it cool down. Stir the jam until smooth. Brush half of the cookies with jam. Place one of the remaining biscuits on each coated biscuit.
Mix icing sugar and rum with the whisk of the hand mixer until smooth. Spread the top of the cookies with the whisk and let them dry.