In a bowl, using the back of a wooden spoon, crush together the mint sprigs, sugar and lime juice. Pour in the rum, cover with plastic wrap, and let stand at room temperature for at least 2 hours or up to overnight.

Select 4 tall, slender glasses. Spread a layer of sugar on a small, flat plate. Working with 1 glass at a time, run a lime wedge around the edge of the glass to moisten it, then dip the rim into the sugar to coat it evenly. Put the glasses in the freezer to chill for at least 15 minutes.

When ready to make the mojitos, fill a blender halfway with ice cubes. Pour the rum mixture through a fine-mesh sieve into the blender. Add the club soda and the 1/2 cup mint leaves and puree until well blended.

Divide the mixture evenly among the chilled glasses. Garnish each glass with a lime wedge and a mint sprig. Serves 4.