Place sugar and water in a saucepan, and stir over low heat to dissolve sugar. Add mint and ginger. Bring to simmer, simmer for 10 minutes. Set aside to cool. Strain into large bowl, pressing down on ginger and mint to extract as much flavour as possible. Discard mint and ginger.
Drop the Broccolini® into a saucepan of boiling water. Cook 2 minutes or until bright green and stalks are just tender. Drain. Refresh in cold water. Pat dry with paper towel then roughly chop. Put into a blender with the coconut water. Blend until well combined. Stir into the ginger syrup. Pour mixture into a shallow container and freeze 6 hours or until almost frozen, still icy in the centre.
Break up the frozen sorbet with a metal spoon and transfer to a food processor. Pulse until the sorbet is smooth but still icy. Add the egg white, pulse for 20 seconds. It is important not to over process the sorbet or it will melt.
Spoon the sorbet into a container just large enough to contain it. Cover and freeze for 4-6 hours or overnight until firm. Remove from freezer 5 minutes before serving to soften slightly. Serve immediately.