How to Make It
In a food processor, pulse the flour with the sugar. Add the butter and pulse until the mixture resembles coarse meal. Beat the egg with the vanilla and add to the processor; pulse just until the dough begins to come together.
Scrape the dough out onto a work surface and knead briefly. Pat into a disk, wrap in plastic and refrigerate until chilled, about 1 hour or overnight.
Preheat the oven to 350°. On a lightly floured work surface, roll out the dough to an 11-inch round. Fit the dough into a 9-inch springform pan, gently pressing it into the bottom of the pan and 1 inch up the sides without stretching. Prick all over with a fork and freeze until firm, about 10 minutes. Bake the crust for about 15 minutes, or until golden brown. Transfer to a rack to cool.
Reduce the oven temperature to 325°. In a large bowl, using an electric mixer, beat the cream cheese with the mascarpone and sugar at moderately high speed until light and fluffy, about 5 minutes. Add the vanilla extract and the seeds from the vanilla bean and beat until blended. Add the eggs, 1 at a time, beating well after each addition.
Pour the mixture on top of the cooked crust in the springform pan and bake in the center of the oven for about 45 minutes, or until lightly browned on top and still slightly jiggly in the center. Transfer the cheesecake to a rack and let cool completely, then cover with plastic wrap and refrigerate overnight.
Run a thin knife around the edge of the cheesecake to loosen it, then remove the side of the pan. Cut the cake into wedges and serve with the Lavender Syrup and Hazelnut Tuiles.