Slice the half cucumber into rounds and reserve for garnish.

If you have a juicer, juice the whole cucumber, unpeeled. If you don’t have a juicer, quarter the cucumbers lengthwise, scrape out the seeds with a spoon and discard the seeds. Coarsely chop the cucumber and puree in a food processor until smooth. Strain through a fine sieve, pressing on solids to extract as much liquid as possible. You should end up with approximately 1 cup plus 2 Tablespoons of cucumber juice. If you end up with more than that, reserve for another use or add to the cocktail, if desired.

Place mint and lime juice in a pitcher and press on the mint leaves with a wooden spoon or muddler to extract the flavor from the mint leaves.* Add the 1 cup + 2 Tablespoons cucumber juice, lime juice, vodka, Cointreau, and apple juice and stir to combine.

Pour into 4 short glasses filled with ice. Garnish the glasses with cucumber slices and/or mint leaves.