Peel and chop one cucumber. Add the pieces to a blender and blend until smooth.
Strain the cucumber juice through a fine-mesh strainer into a bowl. This removes excess pulp.
Place mint leaves in a tall glass. Use a muddler to gently press the leaves to release their oils.
Add 4 tablespoons of the strained cucumber juice, lime juice, and simple syrup to the glass. Stir well to combine.
Fill the glass with ice cubes. Top with club soda and stir gently.
Garnish with a mint sprig and cucumber slice before serving.
Notes
Store extra cucumber juice in an airtight container for up to 2 days.For best results, use fresh mint leaves and English cucumbers.To make simple syrup, heat equal parts water and sugar until dissolved. Let cool before using.For a sweeter drink, add more simple syrup. For less sweetness, reduce or skip it.This recipe makes one serving. Double or triple ingredients to serve more people.