Combine water and granulated sugar in a small saucepan.  Bring to a boil, then reduce heat to low and simmer, without stirring, for 2 minutes. Remove from heat.

Add the diced peaches to the bowl of a food processor and purée until smooth, about 30 seconds.

Add the sugar syrup, lemon juice, and vanilla bean paste and purée for 5-10 seconds more.

Transfer the peach purée to a shallow baking tin or plastic food storage container. (Mine measures approximately 6 x 9 inches.)

Freeze for 1 1/2 hours. Remove from freezer and use a fork to scrape the outer edges, stirring the partially frozen pieces back into the mixture.

Return to freezer and freezer for 30 minutes more. Stir again.

Continue stirring every 30 minutes, until the mixture resembles finely crushed ice. The entire process will take several hours, depending on the depth of your container and the temperature of your freezer. From start to finish, my peach granita required about six hours of freezing and stirring to reach the desired consistency.

Scoop into glasses. Serve immediately.

Store leftover granita in the freezer for up to one week. Allow leftover granita to sit at room temperature for a few minutes, then break apart with a fork and enjoy.