In a small bowl, combine sugar with 1/2 cup warm water to make a simple syrup. Stir until sugar dissolves completely; let it cool.

In a large pitcher, add mint leaves and lime wedges. Use a wooden spoon or muddler to gently crush them—just enough to release the oils without shredding the mint.

Pour in the cooled simple syrup and white rum. Stir well so the flavors meld evenly.

Add ice cubes to fill the pitcher halfway, then top off with club soda. Give it a gentle stir to distribute the bubbles.

Garnish each glass with a mint sprig and a lime wedge. Serve immediately for maximum fizz.