In a pitcher, combine the Simple Syrup, vodka, lime juice, and club soda. Pour into ice-filled glasses and garnish with mint sprigs.
Mint Simple Syrup:
In a small saucepan, over medium heat, combine the sugar, water, and mint leaves. Bring to a boil, reduce the heat, and simmer for 5 minutes, stirring occasionally, until the sugar has dissolved. Remove the pan from the heat and allow the syrup to cool for 20 minutes. Strain before using, pressing on the mint leaves to extract as much syrup as possible.