In a bowl, combine the espresso and 1/3 cup of sugar, and stir until the sugar is dissolved. Pour into a shallow baking dish and freeze until ice granules start to form around the edges, about 1 hour. With a fork, stir and break up the granules. Freeze and stir every 20 minutes until the mixture becomes an icy slush, about 2 hours. Cover and keep frozen until ready to serve.
In a medium bowl, whip the cream until soft peaks start to form. Add the remaining tablespoon of sugar and the liqueur and beat until just stiff.
To serve, scoop the granita into shallow wide-mouthed glasses or ice cream coups and top each serving with the sweetened cream and chocolate curls.