Place the sugar and tarragon in a food processor or blender and pulse until the tarragon is finely ground and the sugar is green. Transfer the tarragon sugar to a medium bowl; add the apple juice and lemon juice and whisk to combine. Let the mixture stand for 10 minutes, stirring it occasionally, until the sugar has dissolved.
Pour the mixture into a 9-inch-square nonreactive metal baking pan and place it in the freezer. Stir the mixture, pulling a fork back and forth through it, every 45 minutes or so for 4 hours, or until frozen. The mixture will become a fluffy, icy concoction.
To serve the granita, spoon it into chilled dishes.