Step 1
Make the mulled cider syrup: In a small pot, bring the cider to a boil over medium-high heat. Turn the heat to medium and cook, stirring occasionally, until the cider is reduced by half, about 2 minutes. Add the nutmeg, cloves, cardamom, cinnamon, and star anise, then remove from the heat and set aside to cool. Strain the syrup into an airtight container and refrigerate. (You should have about ½ cup of syrup.)
Step 2
Make the cocktail: To a cocktail shaker filled halfway with ice, add the vodka and 2½ ounces of the mulled cider syrup. Shake until chilled, then strain into two champagne glasses, top each with prosecco, and serve.