Prep. Fill two drinking glasses with ice and a few cucumber slices, and set aside.
Make the simple syrup. Combine the water and sugar in a small saucepan and bring to a boil. Stirring occasionally, continue to boil for 5 minutes, or until the sugar is dissolved. Remove from heat, set aside, and let cool.
Muddle. Add 10 cucumber slices, mint leaves, 1 to 2 ounces of simple syrup, and lime juice to a cocktail shaker (or a mason jar). Muddle well until the cucumber is macerated and the mint is broken down.
Shake. Pour in the rum, cover tightly, and shake vigorously to combine.
Serve. Strain the cocktail into the glasses, top with soda water, garnish with a fresh mint sprig, and serve.
Notes
Store the remaining Simple Syrup covered and in the refrigerator for up to 1 month.