In a large bowl, combine coffee, 1/2 cup sugar and liqueur, if using. Stir until sugar is completely dissolved. Pour into a 9-by-13-inch baking pan, cover with foil or plastic wrap and freeze until mixture is solid, about 4 hours. (It may be slightly slushy in spots.)
In a bowl, beat cream and remaining 1 Tbsp. sugar until soft peaks form. Using a fork, scrape coffee mixture into fine crystals, breaking up large chunks. Spoon granita into 6 dessert glasses or small bowls and top each serving with a dollop of whipped cream.