Place the strawberries in a blender or food processor and puree until smooth. Pass the strawberry puree through a fine sieve into a medium bowl.
In a small saucepan, combine the water with the sugar and cook over moderate heat, stirring, just until the sugar has dissolved. Let the sugar syrup cool. Add the sugar syrup to the strawberry puree along with the lime juice. Transfer the granita mixture to a large, shallow nonreactive roasting pan and freeze just until ice crystals begin to form around the edge of the pan, about 30 minutes. Using a fork, scrape the ice crystals into the center of the roasting pan. Continue to freeze the granita for about 2 hours, scraping the ice crystals into the center every 30 minutes, until the granita is fluffy and icy. Transfer the strawberry granita to bowls and serve.
The strawberry granita can be made up to 2 days ahead. Let it stand at room temperature for 15 minutes, then scrape the granita until it is fluffy and icy before serving.
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