Step 1Place the cucumber and gin in a clean glass jar. Seal the lid tightly and set aside for 5 days, shaking once a day. Strain the gin through a few layers of cheesecloth or a fine mesh strainer into the original gin bottle. Label the bottle and refrigerate before using.
Step 1In a medium saucepan, combine the water and sugar over high heat. Boil for 10 minutes, stirring occasionally until the sugar is completely dissolved. Remove from heat and add the ginger, allowing it to steep until the mixture is fully cool. Strain the syrup through cheesecloth or a fine mesh strainer into a clean glass jar. This makes about 1 cup ginger simple syrup, more than is needed for a single cocktail. Cover and refrigerate; the syrup will keep, refrigerated, for 2 weeks.
Step 1In the bottom of a cocktail shaker, muddle together 5 basil leaves with the lime juice. Fill the shaker with ice, then add the gin and simple syrup. Shake the ingredients vigorously, then strain into a chilled cocktail glass. Garnish with the remaining basil leaf and serve immediately.