Place the mint sprigs in a glass with a lime wedge.
Use a muddler (or the bottom of a wooden spoon) to crush the mint and lime in order to release the mint oils and lime juice.
Add 2 more lime wedges and the sugar, then muddle again to release more lime juice. Do not strain the mixture.
Fill the glass almost to the top with ice.
Pour the rum over the ice, and fill the glass with club soda.
Stir, taste, and add more sugar if desired.
Garnish with some thin lime slices and a sprig of mint. Serve immediately. Enjoy.