Make the mint simple syrup first. Combine the sugar and half cup of water in a small saucepan over medium heat. Stir until the sugar dissolves completely, about two minutes. Remove from heat, add the mint leaves, and let them steep for ten minutes. Strain out the mint and set the syrup aside to cool slightly.

Peel the cucumber and cut it into rough chunks. Add the chunks to the blender with one cup of cold water. Blend on high for about 30 seconds until the cucumber is completely liquefied and the mixture looks like a thin green slurry.

Pour the blended cucumber through a fine mesh strainer into a large pitcher, pressing on the solids with the back of a spoon to extract as much liquid as possible. Discard the pulp or save it for compost.

Add the fresh lemon juice and the cooled mint simple syrup to the pitcher with the cucumber juice. Stir well to combine. Taste at this point. If you want it sweeter, add another tablespoon of simple syrup. If you want more tartness, squeeze in a little extra lemon.

Add the remaining two cups of cold water to the pitcher and stir again. The lemonade should be pale green, lightly sweet, and bright with lemon. Refrigerate until very cold, at least 20 minutes, or serve immediately over a full glass of ice. Garnish each glass with a fresh mint sprig.