Make the mint simple syrup: stir together the mint leaves, sugar and water in a small saucepan. Bring to a boil, stirring until the sugar dissolves. Reduce heat and simmer for 20 minutes. Allow the syrup to cool to room temperature. Strain out the mint leaves and pour into an airtight jar or container. Store in the refrigerator.
Pour the bourbon and mint simple syrup into a silver cup or other insulated mug and stir gently. Pack the glass with ice and top with a few dashes of bitters. Garnish with extra sprigs of mint.