Add lime wedges and sugar to a sturdy glass or pitcher, then muddle gently for about 1 minute until the limes release juice and the sugar begins dissolving.
Add fresh mint leaves and lightly press them to release aromatic oils without shredding the leaves completely, which helps maintain a smooth flavor.
Pour in fresh lime juice and cold water, then stir thoroughly until the sugar dissolves evenly throughout the mixture.
Fill serving glasses almost completely with ice cubes because plenty of ice keeps the drink cold and slows dilution.
Pour chilled club soda into each glass and stir gently so the drink stays fizzy and refreshing instead of losing carbonation too quickly.
Garnish with fresh mint leaves and lime slices, then serve immediately while the bubbles and mint aroma are at their freshest.