Heat the butter and sugar beet syrup together in a small saucepan over medium heat, stirring, until the butter is completely melted, 2-3 minutes. Remove from heat and let cool to room temperature, about 10 minutes.
In a bowl, combine the flour, baking soda, cinnamon, ginger, cardamom, and salt. Stir in the sugar, the cooled butter mixture, and egg with a wooden spoon until combined. On a lightly floured work surface, knead the dough for 1-2 minutes. Shape into a flat, round disc, wrap in cling film, and refrigerate for 30 minutes.
While the dough chills, preheat the oven to 180°C (356°F). Line two baking sheets with parchment paper.
Roll out the dough on a lightly floured work surface to 5 mm thickness. Cut out stars using any size star-shaped cutter. Arrange on the prepared baking sheets and bake for 10 minutes until lightly golden. Do not overbake or they will dry out. Transfer to a wire rack with a spatula and let cool completely.
Mix the powdered sugar with roughly a little milk to reach a spreadable consistency. Decorate the cooled stars with icing as desired.