In a small saucepan, heat 1/8 cup of water with the 1/4 cup of granulated sugar over medium high heat. Bring to a boil, stirring occasionally until the sugar is dissolved, about 3-5 minutes. Remove from heat and stir in the fresh mint leaves. Let the syrup steep for 1 hour.
Cut the English Cucumber into large chunks and throw it through a juicer. If you do not have a juicer, you can puree the cucumber in a blender or food processor, then push the pulp through a fine mesh strainer into a bowl.
In a cocktail shaker filled with ice, pour in the cucumber juice, gin, lime juice and 2 ounces Minty Simple Syrup. Shake vigorously and pour over ice with a cucumber round float.