In a shaker, combine Branca Menta, vodka and apple brown sugar syrup with ice. Shake vigorously, then strain into glass over ice. Top with iced tea and garnish with fresh apple chunks and mint sprigs.
To make apple brown sugar syrup:
Combine all ingredients in a medium saucepan. Turn heat to medium and bring to a boil. Reduce heat and simmer for 40 minutes, or until the syrup has thickened. Strain into a glass jar to cool. Discard apple peels.