Place sugar, water, lemongrass and lime leaves into the Thermomix bowl and heat 8 min/90°C/speed 3.
Allow to infuse in Thermomix bowl for at least 30 minutes before pouring the contents into a nut milk bag and discarding solids. (I think longer is better.)
Return liquid to the bowl and add lime oil, lime juice. Pour into a tray and freeze overnight. Due to the fairly high sugar content it will not freeze rock hard.
When ready to serve, place chunks of the frozen mixture into the Thermomix bowl and blend 10 sec/speed 8 or until slushie. Top up with sparkling mineral water to dilute to taste. Serve immediately garnished with either a lime leave or some sliced lime. (Big straws are a must!)