Add several mint leaves (no stems) and several cucumber slices into a shaker tin, along with the lime and simple syrup, then muddle for several seconds.
Add the gin and a generous amount of ice.
Shake vigorously and strain with a Hawthorn strainer into a chilled coupe and garnish with a thin cucumber slice (if it is too thick, it will not float on top, but you can also slice the cucumber to sit on the side of the glass).