Cut each of the lime halves into four chunks. Put in a large mortar with the sugar and squish well with a pestle to break them down – they will release their juices, leaving you with a sweet lime syrup. Remove the lime peel, squeeze to release the last bits of juice, then discard. Transfer the lime syrup to a small plastic container and freeze for at least 4 hrs, ideally overnight.
When you’re ready to make the cocktail, put the ice in a blender suitable for crushing ice, then add the frozen syrup and cachaça. Blend at a high speed until smooth and thick. Transfer to cocktail glasses or tumblers and serve immediately.