In a sturdy, heavy-bottomed, tall cocktail glass, add the sugar and sprigs of  mint 

Place the mint and the sugar in a tall sturdy glass or use a cocktail shaker if you think the serving glass is too fragile.

Muddle the mint with the sugar about 5 to 10 times, until the mint is very fragrant.

Squeeze the juice of one lime into the mixture and muddle the remaining lime shells with the mint and sugar mixture, 5-10 times to release the oil from the lime skins.

Pour in the rum and the club soda, and gently stir a few times to help dissolve the sugar.

You can strain the mixture at this point, however the traditional way is to leave it unstrained.  I do scoop out the lime peels usually, but you can leave them in.

Gently stir to combine, then use your spoon to drag some of the muddled mint higher up into the glass.

Taste, and stir in more sugar or lime if desired. 

Use the remaining lime cut in wedges and some mint sprigs to garnish