Put the sugar in a pan with 400ml water and warm over a low heat until dissolved. Increase the heat and boil for 1 min. Remove from the heat and allow to cool slightly.
Pour the tonic water into a bowl, add the sugar syrup, then chill in the fridge until cold.
Once the tonic mix is cold, add the gin, lime and lemonzests and juice, and pour into a container, then freeze until just frozen – this will take about 2 hrs.
Remove the just-frozen sorbet from the freezer and break up with a fork. Froth the egg white with a fork, then add both to a food processor and pulse to blitz together – do it quickly so it doesn’t actually start to melt. Pour the mix back into your container and freeze until solid. Freeze extra slices of lemons and limes on a tray until solid. Serve the sorbet scooped into glasses or small bowls topped with lemon and lime slices