Make the lemon granita: In a bowl, combine all the granita ingredients. Stir until the sugar is dissolved. Freeze in a small, shallow container (about 5x7x2 inches) until the mixture is just slushy, about 1-1/2 hours. Beat the mixture with a fork and return to the freezer until it's completely frozen, another 2 to 3 hours at least. Keep frozen until ready to serve.

Make the compote: Put the strawberries and the 1/4 cup sugar in a stainless-steel bowl that fits over a saucepan to make a double boiler. Cover with plastic wrap and cook over a low flame until the strawberries have given up their juices, about 45 minutes. Remove from the heat and transfer the berries to a colander set over a bowl to catch the juices. Put both the colander and the bowl in the refrigerator to drain overnight, or until the strawberries are cold, 5 to 6 hours.

Pour the strained strawberry juices into a saucepan or a microwave-proof bowl (you'll have about 3/4 cup). Sprinkle the gelatin onto the cold strawberry juices and let the granules swell and soften, about 3 minutes. Heat the gelatin mixture to melt it, about 2 minutes on medium heat or 15 to 45 seconds in the microwave. In another bowl, combine the lemon juice, strawberry puree, and the remaining 2 tablespoons sugar. Stir in the gelatin mixture, and let sit until it has the consistency of unbeaten egg whites, about 1 hour at room temperature (to speed it up, put the bowl over an ice bath or in the fridge). Add the drained, cooked strawberries, stir gently, and immediately distribute the mixture evenly among the eight glasses you're using. Refrigerate until it's set.

Meanwhile, make the cream cheese mousse. Spoon equal amounts of the mousse over the compote in the glasses. Chill until needed, at least 30 minutes.

Prepare the strawberries with lemon and basil: Put the strawberries in a bowl. Add the remaining ingredients; mix gently. Chill until ready to serve.

Assemble the parfaits: Just before serving, remove the glasses with the compote and mousse from the refrigerator. Break the granita into large chunks, put it in a food processor, and pulse just until it's crushed and snowy. Spoon a layer of seasoned strawberries over the mousse and sprinkle some of the granita over the berries. Garnish each with a fresh strawberry and a basil leaf. Serve right away.

Make Ahead Tips: The granita can be made up to a week ahead and the compote up to two days ahead, but the strawberries with lemon and basil must be done at the last minute.