In a sturdy glass or cocktail shaker, add the fresh mint leaves and granulated sugar. Use a muddler to gently press the mint leaves against the bottom of the glass. This releases the essential oils from the mint, which is crucial for flavor.

Cut the lime into quarters. Squeeze the juice from the lime quarters into the glass, ensuring to catch any seeds. Add the lime wedges into the glass as well for additional flavor.

Add the white rum to the glass with the muddled mint and lime. Stir gently to combine the ingredients, ensuring the sugar dissolves completely.

Fill the glass with crushed ice, packing it tightly to keep the drink cold.

Top the glass with chilled club soda, pouring gently to avoid splashing. Stir the mixture lightly to combine.

Garnish with a sprig of fresh mint and a lime wedge on the rim of the glass for presentation.

Serve immediately with a straw or a spoon for stirring.