Green Goddess Cocktail

Combine vodka and tea in a jar. Cover and let infuse 8 hours. Strain through a fine-mesh sieve into a small bowl; discard solids. Cover and chill. Bring sugar and 1/3 cup water to a boil in a small saucepan, stirring to dissolve sugar. Remove from heat, add arugula, and stir until wilted. Cover and let syrup steep 5 minutes, then strain into a small bowl; discard solids. Cover and chill arugula syrup until cold, at least 1 hour. Cut 12 thin slices from cucumber; set aside. For each cocktail, finely grate onefourth of remaining cucumber directly into a cocktail shaker. Add a piece of jalapeño and muddle until chile is broken into small pieces. Add 2 ounces green-tea vodka, 1 ounces arugula syrup, and 1 ounces lemon juice to shaker. Fill with ice and shake vigorously until outside of shaker is frosty, about 30 seconds. Strain cocktail into a Collins glass filled with ice and garnish with mint and 3 reserved cucumber slices.