Slice the length of the cucumber to get a slice that you can curl for the garnish. Curl and stab with a cocktail skewer and sit aside.
Chop up the balance of the cucumber. Add cucumber to cocktail shaker and muddle. If you don’t have a muddler, use a wooden spoon.
Cut lime in half. Then slice a garnish from one half. Juice the rest into the cocktail shaker. Add the gin and cointreau into the shaker, and add ice. Shake.
Strain cocktail into a martini glass and garnish with cucumber slice on skewer.