Place sugar and mint into Thermomix bowl and blend 10 sec/speed 10.
Add limes and about 500g water as well as some ice. Chop 3 sec/speed 6. Mix 10 sec/Reverse/speed 2.
Strain this mixture into serving jug with ice and more soda water. Stir through the white rum to taste and serve.
You can remix the residual sugar and lime mix by adding another 500g water and repeating the whole process. It will not be as strong, but will yield at least another round!
Serve over ice and dilute to taste with any additional club soda and rum as you wish! Garnish with fresh mint leaves.