How to Make It
In a small bowl, sprinkle the gelatin over 1 tablespoon of water. In a medium bowl, whisk the egg yolks with the sugar.
In a medium, heavy saucepan, combine the cream and vanilla and bring to a simmer. While whisking constantly, slowly drizzle the cream into the egg yolk mixture. Pour the custard base into the saucepan and cook over moderately low heat, stirring constantly, until thickened enough to coat the back of a wooden spoon, 3 to 4 minutes. Strain the custard into a medium bowl. Add the softened gelatin and whisk until dissolved. Press a sheet of plastic wrap directly onto the surface of the pudding and let cool to room temperature.
In a blender, puree the preserved lemon and lemon juice until well combined. Stir the lemon puree into the pudding. Cover again with plastic wrap and refrigerate the pudding until cold and thick, at least 3 hours or overnight.
Fill a medium bowl with ice water. In a medium saucepan of boiling water, blanch the 1 cup of basil leaves until they are very tender, about 3 minutes. Drain and transfer the basil to the ice bath to cool. Drain well and squeeze dry. In a blender, combine the basil and simple syrup with 1 tablespoon of water and puree until smooth.
Spoon the preserved-lemon pudding into bowls. Drizzle with the basil syrup, garnish with basil leaves and serve.
To make the simple syrup: In a small saucepan, combine 1/4 cup sugar and 1/4 cup water and cook over low heat, stirring, until the sugar dissolves. Refrigerate the syrup until chilled.