Use a vegetable peeler to peel the outer layer of the lemons, creating several strips of lemon zest. Use a juicer to extract juice from the peeled lemons.
Combine sugar, water, and 1 cup lemon juice (supplementing with additional lemon juice as needed to equal one cup) in a saucepan over medium heat. Stir and cook until this mixture comes to a simmer and the sugar disolves. Remove from heat and add the basil leaves and lemon zest strips. Allow the basil to seep for 5 - 10 minutes.
Strain the lemon basil syrup, to remove the basil and zest pieces. Pour this into a mason jar or lidded container and refrigerate until thoroughly chilled (at least one hour).
When you're ready to serve the lemonade, add the water or club soda, crushed ice and basil sprigs to a pitcher with the lemonade concentrate. Pour into individual glasses. Garnish with fresh basil leaves and lemon slices.