To make the agave, bring agave nectar to a simmer over low heat in a small saucepan.
With the blunt edge of a knife, bruise lemongrass (pound it) to bring out the fragrance and add it to the saucepan.
Allow lemongrass to simmer for 3-5 minutes, then remove from heat and immediately transfer to a non-reactive (heat-resistant plastic or other non-metal) container.
Let agave nectar rest with lemongrass stalks inside at room temperature for at least 6 hours. After infusing, remove lemongrass and discard.
Refrigerate agave until ready for use.
To make the cocktails, combine lemon juice, agave nectar and basil leaves in a glass and muddle together to blend flavors.
Add muddled mixture to a cocktail shaker with ice and gin.
Top with a splash of soda water.
Close cocktail shaker tightly and shake vigorously.
Strain cocktail and serve immediately over ice.