Remove 2 wide strips zest from the lime with a vegetable peeler; set aside. Cut the lime in half; thinly slice one half and cut the remaining half into 4 wedges for garnish.
Bring 1/3 cup water and the sugar to a boil in a saucepan over medium heat, stirring, until the sugar dissolves. Remove from the heat; add 1/2 cup each basil and mint and the lime slices and let cool completely.
Transfer the herb syrup to a blender and blend until the herbs are finely chopped (the limes will still be mostly intact). Strain through a fine-mesh sieve into a bowl. Freeze until slushy, about 2 hours.
Combine the cachaça, limeade concentrate, lime zest, the remaining 1/2 cup each basil and mint and the frozen herb syrup in a blender. Add 5 cups ice and blend until thick and slushy; gradually add up to 1 more cup ice if needed. Pour into glasses and garnish each with a lime wedge.
Photograph by Levi Brown