Place sugar, water and mint in a saucepan and heat gently, stirring occasionally, to dissolve the sugar, until it reaches boiling point and the sugar has dissolved completely. Let cool.
While the syrup is cooling, cut up the cucumbers and purée.
Add the cooled syrup to the cucumber purée and blend until it’s a nice, smooth consistency. Strain out the pulpy bits using a sieve.
Whisk egg white until it’s foamy, but not stiff. Add to the cucumber mint mixture.
Put in a freezer safe bowl and into the freezer. Check on it regularly and give it a good stir as it freezes.
Before serving, let it sit out for a few minutes to soften up and becomes scoopable.