In a food processor, combine the butter, sugar, vanilla sugar, rum, cardamom, cinnamon, flour, baking powder, and salt. Process until a dough forms. Divide the dough in half and shape each half into a roll approximately 3 cm in diameter. Wrap each roll tightly in plastic wrap and refrigerate for at least 8 hours or overnight.
Melt the milk couverture in a water bath. Set aside to cool.
Preheat the oven to 175 °C / 347 °F using top and bottom heat (conventional setting).
Cut each dough roll into slices 1.5–2 cm thick. Arrange the slices on a baking sheet with a little space between each. Bake on the middle rack for 10-15 minutes until light golden.
Cool the cookies on the baking sheet for 10 minutes, then transfer to a wire rack and let cool completely, about 20 minutes. Gently reheat the couverture if it has solidified, then dip the top half of each cookie into the couverture and place on parchment paper. Let the chocolate coating set for 20-30 minutes at room temperature. Store in an airtight container; the cookies taste best after 1-2 days.