Scrape the thin skins off the ginger with a sharp knife. Using a granite pestle or the blade of a heavy cleaver, bruise the ginger well. Place ginger in a small pan with the water and bring to the boil. Turn down heat and simmer for 10 minutes. Strain ginger juice and cool completely.
In the jug of an electric blender or food processor, combine the cold ginger juice, sugar, yoghurt, ice cubes and mint leaves. Whiz until smooth. Top up with the 7-up and and serve immediately.