Peel cucumber with a vegetable peeler, then dice into ¾-inch cubes
Scrub lemons and zest them with a microplane, then cut in half for juicing
In a high-speed blender, combine cucumber, 1 cup water, and half the lemon juice
Blend on high for 30 seconds until completely liquid with no visible pulp
Strain the mixture through a fine sieve into a pitcher, pressing on fibers to extract more liquid
Add remaining 3 cups water to the pitcher
Squeeze in the rest of the lemon juice from the zested halves
Stir in the sugar/honey and let sit at room temperature for 5 minutes to bind flavors
Place a fine sieve over a clean pitcher and churn the mixture through it for a smooth texture
Chill lemonade for at least 1 hour before serving
Garnish each glass with mint sprigs and serve