Place 8 mint leaves and the granulated sugar in a chilled 12-ounce highball glass. Add the freshly squeezed lime juice and gently muddle 10–15 seconds until the leaves are bruised and fragrant, not shredded.
Pour in the white rum and stir 10–15 seconds until the sugar is mostly dissolved.
Fill the glass with crushed ice (about 1 cup). Top with the chilled club soda (2 oz) and briefly stir 5 seconds to lift the mint through the drink.
Lightly clap the mint sprig between your palms to release aroma, then garnish the drink with the mint sprig and a lime wheel. Serve immediately; the drink should be cold, aromatic, and lightly effervescent.