Muddle ginger, cucumber, and lime in a bowl. Strain the mixture into a measuring cup using a small-grid strainer. Use the back of a spoon to extract all the juice.

Clap the mint leaves together in your hands 3 or 4 times to release the oils, then divide them between two glasses. Divide the ginger, cucumber, and lime juice into the two glasses. Add ice and top each glass with half the ginger ale and sparkling water. Garnish each with a cucumber slice and mint sprig.