Make the mint syrup: Bring the sugar and 1 cup water to a simmer in a small saucepan over medium heat, stirring until the sugar is dissolved. Remove from the heat and add the mint sprigs, pressing with a wooden spoon to submerge them. Let cool to room temperature, then remove the mint with a slotted spoon and discard. Refrigerate the syrup until chilled, at least 1 hour. (You will have about 1 1/2 cups syrup; cover and refrigerate up to 2 weeks.)
Make the drink: Put the mint leaves and 1 ounce of the mint syrup in a small glass or julep cup. Muddle with the handle of a wooden spoon until the leaves start to break apart. Stir in the bourbon. Fill the glass with crushed ice. Garnish with a mint sprig.
To store fresh mint, wrap it in a damp paper towel, then place in a resealable plastic bag and refrigerate for up to a week.