Blend the cucumbers, sugar, lemon juice, mint, salt and 4 cups cold water in a blender until as smooth as possible (a fair amount of solids will remain). (If all the ingredients do not fit in your blender, use half the water to puree, transfer the puree to a larger container, then stir in the remaining water.) Refrigerate to chill, at least 1 hour.
Before serving, strain the cucumber-mint water through a fine strainer. Use the bottom of a ladle or wooden spoon to help push the liquid through the mesh. Discard the solids. Serve over ice. Add gin or vodka to taste if using.
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