Place 1/4 cup of sugar and 4 basil leaves in the blender or food processor and process until well blended. (You can also make the basil sugar using this method.) Pour the sugar onto a plate and set aside.
Put the remaining basil leaves in a large shaker along with the lime juice and a tablespoon or so of the remaining sugar. Muddle this mixture until the basil leaves have broken into little pieces. Add the rum and several ice cubes. Cover and shake until well chilled.
Rub a lime wedge along the rim of two short glasses. Dip in the basil sugar. Add a couple ice cubes to each glass and divide the drink between them. Top off with club soda and stir. Taste and add a bit more sugar, if desired. Garnish with a lime wedge. Enjoy!
* We made Strawberry Basil Mojitos too! Muddle the basil with 1/2 ounce of Strawberry Simple Syrup instead of the tablespoon of white sugar. Add more syrup to taste, after you have filled each glass.