2 tablespoons orange-flavored liqueur (recommended: Grand Marnier)
Clean and wash the oranges. Cut off the top 1/4 of the oranges and shave a small slice off the bottom so it stands easily.
Using a grapefruit knife, cut out the flesh of the oranges and place in a strainer set over a bowl. Be careful not to cut through the bottom. Stand the oranges on a cookie sheet and place in the freezer and freeze until solid.
Press the flesh of the oranges through the strainer to extract all of the juice. You should end up with about 1 1/2 to 1 3/4 cups of orange juice. Add the extra 4 cups orange juice. Stir in sugar, a tablespoon at a time, until the mixture is of desired sweetness. Add the basil and orange-flavored liqueur and let sit for 15 minutes.
Strain the mixture into a roasting pan or baking dish. The dish should be large enough so the liquid is not more than 1-inch deep. Freeze the mixture for 1 hour. Remove from the freezer and scrape with 2 forks to break up ice. Return to the freezer and freeze until solid, about 2 to 3 hours, scraping with forks every hour or so.
When the granite is frozen, scoop it into the orange shells and serve immediately.