Special equipment: a muddler or a spoon
Combine the sugar with 1 cup water in a medium saucepan and bring to a simmer over medium-high heat, stirring until the sugar has dissolved. Turn off the heat and add the mint leaves. Muddle with a muddler or the back of a spoon. Allow to cool completely, about 20 minutes. Strain and discard the mint leaves.
Add the lime juice and 4 cups water to a large pitcher. Pour the simple syrup into the limeade. Stir to combine.
To serve, pour the limeade into glasses filled with ice. Top with some sparkling water. Garnish each with a lime round and mint sprig.