Place 2 of the mint sprigs, the lime juice, and the simple syrup in a cocktail shaker and press gently against the mint with the back of a spoon to release the oils.
Add the rum and a handful of ice and shake vigorously until the mixture is well chilled, about 20 seconds.
Fill a 10-ounce glass with ice and strain the drink into the glass.
Top with club soda and garnish with the remaining mint sprig.