Blend the cucumber with water in a blender until smooth.
Strain the puree through a fine strainer into a glass jar or bowl. Discard the pulp left behind and reserve the juice for the cocktail.
To make the cucumber martini
In a martini shaker, combine ice, gin, cucumber water, simple syrup, lemon juice, and fresh mint. Shake vigorously until cold. Strain into a martini glass, and garnish with a mint leaf and a slice of cucumber, if desired.
Serve and enjoy!
Notes
Recipe tips:
MAKE-AHEAD: Make the cucumber juice in advance, and keep it cool in the refrigerator, for up to 3 days before serving.
SERVING: Chill the martini glass for a few minutes in the freezer or add ice to the martini glass while you make the cocktail.