First, grab a lowball or highball glass and fill it generously with ice. This will keep your drink chilled from the very beginning. Set it aside while you gather your other ingredients. Having everything ready makes the mixing process smooth and effortless.
In a cocktail shaker, combine the fresh lime juice, sugar, and mint leaves. Use a muddler to gently press down on the mint leaves a few times. The goal is to bruise them slightly to release their fragrant oils, not to shred them into tiny pieces.
Next, add the white rum to the shaker along with a handful of ice. Secure the lid tightly and shake vigorously for about 30 seconds. This part is crucial as it chills the mixture thoroughly and combines all the flavors into a harmonious blend.
Double strain the mixture from the shaker into your prepared ice-filled glass. This ensures no small mint pieces end up in your drink, giving it a smooth texture. Top it off with a splash of club soda, garnish with a fresh lime slice and a few mint sprigs, and serve immediately.