Wash the cucumber thoroughly and slice it thinly. Rinse the mint leaves and gently pat them dry. Lightly crush the mint leaves between your fingers to release their aromatic compounds.
In a large pitcher, combine the cucumber slices and mint leaves. Using a wooden spoon, lightly muddle them together to release their flavors without completely crushing them.
In a separate bowl, combine the freshly squeezed lemon juice with honey or your chosen sweetener. Stir vigorously until completely dissolved.
Pour the lemon-sweetener mixture over the muddled cucumber and mint, then add the cold water. Stir gently to combine all ingredients.
Cover the pitcher and refrigerate for at least 30 minutes to allow the flavors to meld.
Optional: Strain the mixture into a clean pitcher if you prefer a smoother texture, or leave the cucumber and mint for continued flavor infusion.
Pour over ice in tall glasses and garnish with fresh mint sprigs and lemon slices before serving.