Slice plums and remove pits. Cut into smaller pieces and put them in a medium, nonreactive saucepan with the water and mint. Cover and cook over medium heat, stirring occasionally, for 8 minutes or until tender. Remove from the heat and stir in the sugar until dissolved. Add vanilla extract. Let cool to room temperature. Remove mint.

Once the mixture has cooled, run through a blender or food processor until well-pureed. Place in an ice cream canister and prepare according to manufacturer’s instructions.

If you don’t have an ice cream maker, pour into a container with at least a 1 quart capacity, ideally with 2-inch sides. Chill until it is lightly frozen.

For sorbet, run again through a blender or food processor, return to container, and allow to freeze.

For granita, take a fork and stir the mixture, breaking up the frozen parts near the edges into smaller chunks and rake them toward the center. Return the dish to the freezer, then check the mixture every 30 minutes afterward, stirring each time and breaking any large chunks into small pieces with a fork, until you have fine crystals of granita. It if freezes too hard, allow it to sit out for a few minutes, then break it up.

Whichever you’ve made, scoop the frozen dessert into cups or small glasses, garnish with more mint, and add whipped cream if you’d like or serve with small cookies.