Make the mint simple syrup: In a small saucepan over medium heat, combine the water, sugar, and roughly chopped mint leaves. Bring to a low boil, then reduce the heat to low and simmer for 5 minutes, until the sugar is fully dissolved and the mint is fragrant. Remove from the heat and let cool for 5-10 minutes. Strain through a fine-mesh sieve. Store remaining mint syrup in the refrigerator for up to 1 month.
In a cocktail shaker, combine the cucumber rounds, lime juice, whole mint leaves, and 1 tablespoon of the mint simple syrup. Muddle for 1-2 minutes, until the cucumber and mint have released their flavors, but are not mushy.
Pour in the vodka and fill the shaker halfway with ice. Shake vigorously for 30 seconds.
Fill a Collins glass with ice and pour in the strained cocktail.
Garnish with the cucumber ribbons.
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