In a medium bowl, use a fork to crush 1/4 cup fresh blackberries until juicy and the fruit has broken up to the consistency of jam. Add 1 cup rosé wine, 1/4 cup plus 2 tablespoons water, and 3 tablespoons granulated sugar to the fruit mixture. Stir together until the granulated sugar has dissolved completely.
Pour the mixture into a shallow baking dish and place in the freezer until partially frozen. Use a fork to scrape and break the mixture into shards and crystals. Return the pan to the freezer to freeze completely. Scrape and toss the granita one more time. Keep frozen until serving (see baker's notes).
For the Basil Whipped Cream
In a liquid measuring bowl, combine 1/2 cup heavy cream and 3 to 5 fresh basil leaves (use more for a stronger flavor). Stir gently, gently crushing the leaves into the cream, before covering with plastic wrap. Refrigerate to let the basil infuse the cream for at least 3 hours, preferably overnight.