In a 3 1/2- to 4-quart pan, combine cranberry juice, ginger, cinnamon sticks, and star anise. Boil over high heat until reduced to 2 cups, about 1 hour.
Let cool about 30 minutes, then pour through a fine strainer into a small pitcher. Chill syrup until cold, at least 1 1/2 hours.
For each serving, pour 2 tablespoons of the cranberry syrup and 1 teaspoon cognac into a Champagne flute (6 to 8 oz.).
