Whirl 2 basil leaves and 1/4 cup sugar in a food processor until well blended; transfer to a plate and set aside.
Put 6 to 8 remaining basil leaves in a large cocktail shaker with a few ice cubes, the remaining 4 tsp. sugar, and lime juice. Muddle the mixture with the back of a thick wooden spoon, keeping your hand over the top to prevent spilling, until basil has broken up.
Add rum and a few more ice cubes to shaker, cover, and shake to blend. Rub a lime wedge along rim of 2 lowball glasses and dip in basil sugar (reserve remaining sugar for more cocktails). Add a few ice cubes to each glass. Remove top from shaker (do not strain), and divide mixture between glasses. Top off each with club soda and stir. Garnish with a lime wedge.