Using a mandoline or sharp knife, cut 2 thin lengthwise slices from the cucumber. Spiral each cucumber slice into a highball glass; set aside. Cut the remaining cucumber into 1-inch pieces and place in a cocktail shaker. Add the lime juice. Using a muddler, muddle the cucumber until lightly mashed. Add enough ice to the shaker to fill it by two-thirds. Add the gin, cover the shaker, and shake vigorously for 30 seconds. Strain the cocktail into the two prepared high ball glasses, dividing it evenly. Top with the tonic water and stir gently to mix. Garnish with the mint and serve.