1. Make the simple syrup: Combine sugar, cardamom, star anise, and 1 cup of water in a 2-qt. saucepan; cook over medium heat until sugar has dissolved. Remove from heat and cool. Makes about 1 cup. Syrup will keep, refrigerated, for up to 2 weeks.
2. Make the cocktail: Combine gin, persimmon juice, lemon juice, syrup, and vermouth in a cocktail shaker filled with ice; shake vigorously and strain into a coupe glass. Garnish with star anise.