In a small saucepan, combine the Prosecco, sugar, juice of 1 lemon, mint, and salt. Bring to a simmer and cook just to melt the sugar, about 2 minutes. Remove from heat and let cool.

When the Prosecco syrup has cooled, put it in a blender with the peaches and remaining lemon juice. Puree until smooth. Pour into a wide, shallow metal pan so the mixture comes about an inch up the sides. Put in the freezer. Once ice crystals begin to form around the sides of the pan, about an hour, scrape the granita with a fork to incorporate the crystals back into the slush. Keep freezing and scraping every half hour to an hour, until the granita is completely frozen in light, fluffy crystals.