Use the vegetable peeler to peel long strips of the cucumber. Peel until you get a strip that is the width and length you want. Keep the strip pretty thin so it will bend easily.
Gently fold the cucumber strip into a loose zigzag, and thread the strip onto a cocktail pick. Set this part of the garnish aside.
Cocktail
Add the lime juice and cucumber slices to the bottom of your cocktail shaker.
Muddle the juice and cucumber together.
Add in the mint leaves and gently muddle a few times more.
Add in the simple syrup and gin.
Fill the shaker with ice and shake about 15-20 times.
Strain the drink into your serving glass.
Rest the cucumber garnish on top of the glass. Add a mint sprig and serve.