Prep the cucumber: Rinse and trim the ends. If the skin is tough or waxed, peel it.
Slice into chunks for easier blending.
Blend the base: Add cucumber, mint leaves, lime juice, 2 tablespoons simple syrup, and 1 cup of the water to a blender. Blend until completely smooth and bright green.
Strain (optional): For a smoother drink, pour the mixture through a fine-mesh sieve or cheesecloth into a pitcher, pressing gently to extract the liquid. If you like a bit of pulp, skip this step.
Adjust and dilute: Stir in the remaining cup of cold water.
Taste and add more syrup if you prefer it sweeter. Add a tiny pinch of salt to sharpen the flavors.
Chill: Refrigerate for at least 20 minutes, or add a handful of ice to the pitcher if serving right away.
Serve: Fill glasses with ice. Pour in the cooler.
Garnish with cucumber slices, lime wheels, and a slap of fresh mint (give the mint a light smack between your palms to release aroma).
Make it sparkling (optional): Replace the second cup of water with chilled sparkling water just before serving. Do not blend or shake with the bubbles.
Cocktail variation (optional): Add 1–2 ounces of chilled gin or vodka to each glass before pouring in the cooler. Stir gently.