Bring a pan of water and sugar to a boil. Let it boil for 2 minutes and then turn it off. Add in the mint and cover the pan and let it simmer for 10 minutes. Remove the mint, and add the lemon juice. Add more sugar or more lemon to taste.
Pour granita mixture in to a Cook & Freeze glass storage container and once at room temperature, store in the freezer with lid for 5 hours at least. Take out the granita and scrape with a fork and return to the freezer. Repeat this process two or three times over the course of freezing.