Put mint leaves and simple syrup in a blender. Pulse 2-3 times to break up the leaves.
Add rum, lime juice, and crushed ice to the blender. Blend until smooth and slushy.
Pour the mixture into glasses, filling them about 3/4 full.
Top each glass with a splash of club soda and stir gently.
Garnish with fresh mint sprigs and lime wedges.
Notes
The drinks will melt quickly, so serve right away.Fresh mint and lime juice make a big difference in taste compared to bottled ingredients.Keep mint leaves in the refrigerator, wrapped in damp paper towels to stay fresh.Make simple syrup ahead by mixing equal parts sugar and hot water until dissolved. Store in the fridge for up to 2 weeks.Pre-chill the glasses in the freezer for 15 minutes before serving to keep drinks colder longer.