In a sturdy, heavy-bottomed tall cocktail glass, add the sugar and 5 sprigs of mint, reserving 1 sprig for garnish. Muddle the mint with sugar 5 to 10 times until very fragrant. Avoid over-muddling. You want the leaves aromatic, not completely broken down.
Pour in the rum and lime juice. Gently stir a few times to help dissolve the sugar.
Fill the glass with ice.
Top off with club soda. Gently stir in a circular motion to combine, then use your spoon to drag some of the muddled mint higher up into the glass.
Taste and add a touch more sugar if desired.
Garnish with the lime round and the reserved mint sprig. Clap it between your palms first to release the oils. Serve cold with a wide straw.