In the bottom of a pint glass, add, in order: basil, cucumber, lime wedges. Add simple syrup and muddle until the basil is bruised, and all the juice is released from the lime. Add gin and St. Germaine, and shake vigorously until well chilled. Strain through a fine mesh strainer into a chilled coupe or martini glass. Spear the basil sprig through the middle of the cucumber slice, and float it on top of the cocktail as garnish.