Cut 4 thin spears from 1 cucumber, and reserve for garnish. Peel remaining cucumbers, and quarter them lengthwise. Scrape out seeds with a spoon, and discard. Coarsely chop cucumbers, then puree in a food processor until smooth. Strain through a fine sieve, pressing on solids to extract as much liquid as possible.
Fill a cocktail shaker with ice. Add mint, sugar, and lime juice, and shake. Add 3/4 cup cucumber juice, the vodka, and Cointreau, and shake. Strain into 4 glasses filled with ice. Garnish each cocktail with a cucumber spear.