Muddle cucumber slices in the bottom of a cocktail shaker.
Add mint leaves, gin, simple syrup, and lime juice. Fill the shaker with ice, and shake vigorously for 20-30 seconds.
Double strain the cocktail into a chilled coupe glass*. Garnish with a cucumber wheel and mint sprig.
Notes
*Place coupe glass in the freezer for five minutes to chill.*Use a fine mesh sieve to strain the cocktail into the glass. "Double strain" indicates that it strains once through the strainer placed on the opening of the shaker, then a second time through the sieve before entering the glass. This is crucial to prevent any cucumber or mint pieces from getting into the cocktail.