Cucumber Cooler

In a blender, puree cucumbers and water. Pour through a fine-mesh sieve set over a large bowl, pressing on solids (you should have about 2 cups juice; refrigerate overnight, if desired). In a pitcher, mash fresh mint leaves with back of a wooden spoon until bruised. Add cucumber juice, lime, club soda, and agave nectar or honey; stir to combine. Fill 4 glasses with ice, top with drink, and serve with thinly sliced cucumber.