Place the cucumber, tequila, lime juice, ginger, sugar and mirin in the bowl of a food processor and process until smooth. Have a taste to see if you need to add something to make it to your taste.
Pour into a shallow metal tin of approx 20 x 30 cm and freeze for one hour.
Using a fork, rake the top of the granita and freeze for a further 1 hour. Repeat every hour for 3-4 hours or until granita is set. Add the mint and mix gently to combine. Divide over 4 small glasses and serve immediately
Notes
If you cannot find mirin, the original recipe used rice wine vinegar which I did not have on hand, but you can use that if you want