Combine the sugar and 1/2 cup water in a small saucepan. Heat over medium, stirring, until the sugar dissolves, about 2 minutes. Do not let it boil hard.
Remove from the heat, add the mint leaves, press them under the syrup and leave to steep for 15 minutes.
Meanwhile, roll the lemons firmly on the counter and squeeze them. Strain out the seeds.
Strain the mint syrup into a pitcher, pressing the leaves to extract the flavor, and discard them.
Add the lemon juice and cold water, stir and taste. Adjust with more water or a little more syrup, then chill. Serve over ice with lemon slices and mint sprigs.
