To make the sugar syrup, combine 3/4 cup water and 4 tablespoons sugar in a small saucepan. Cook over medium heat, stirring until sugar has dissolved, about 2 minutes. Cool to room temperature.
Place half the strawberries in a blender with the sugar syrup, lemon juice, and vodka, if using. Puree. Transfer to a fine mesh sieve and press on puree, pushing the pulp through. Discard seeds remaining in sieve.
Pour mixture into a shallow 8-inch square pan. Freeze 2 to 3 hours, or until almost completely frozen. Scrape granita with a fork to make it flaky. Freeze for another hour and scrape again. When ready to serve, scrape again.
About 30 minutes before serving, quarter remaining strawberries lengthwise and transfer to a bowl. Add remaining tablespoon sugar, pepper, and vinegar; mash slightly with a fork, keeping mixture chunky. Add basil.
To serve, scrape some granita into four glasses and top with smashed strawberries.