For Orange-Cardamom Gastrique:
Add lemon juice, sugar, and water in a saucepan and cook over medium heat until liquid turns brown and no longer bubbles. Slowly stir in vinegar and cardamom, and gradually add orange juice. Continue to cook over medium heat, stirring, until gastrique is smooth and syrupy. Cool and strain, then add a splash of vodka. Keep refrigerated.
To Assemble and Serve:
Shake brandy and Orange-Cardamom Gastrique with ice and strain into a rocks glass over ice. Garnish with orange twist.