Note: To make cucumber juice, use a juice extractor. Alternatively, peel and chop ½ large cucumber and place in a blender or a food processor fitted with a metal blade. Puree, then strain through a fine-mesh strainer.
Total time: 20 minutes (plus 1 to 3 days for infusing vodka)
Place simple syrup and basil in a cocktail shaker and muddle the basil, gently pressing and twisting to release its oils.
Add vodka, cucumber juice, and lime juice and fill shaker with ice. Shake vigorously until very cold and strain into 2 chilled martini glasses. Garnish with cucumber slices and serve.