Special equipment: Four 3 1/2-inch tart or pastry molds
For the crust: Combine the flour, butter and salt in a food processor, and pulse until the mixture looks like Parmesan. Drizzle in half the ice water and pulse until the mixture forms a rough ball, adding the remaining water if needed. Remove and form the dough into a disk, and then wrap in plastic and refrigerate at least 1 hour.
For the filling: Melt the butter in a small saucepan and add the vanilla bean. Cook until the butter starts to brown and begins to smell like hazelnuts, and then remove the vanilla bean pod and transfer the butter to the bowl of a stand mixer equipped with a paddle. Add the sugar and beat on medium speed. Gradually add the flour. Once the flour has been incorporated, add the eggs one at a time.
For the apples: Melt the butter in a large saute pan over medium heat. Cook the raisins, sugar and apples until the apples begin to soften, 6 to7 minutes. Remove from the heat and pour in the brandy, then return to the fire and flambe, letting the alcohol burn off.
For assembly: Preheat the oven to 350 degrees F. Remove the dough from the fridge 15 minutes before using, allowing it to soften and come to room temperature. Dust a clean work surface with flour. Roll the dough 1/8-inch-thick and lay it into the tart molds with lots of overhang on the sides. Schmear half the filling into the bottom of the tart pans and top with the apples. Fold the extra dough over the top of the apples, and then brush the dough with the egg wash. Bake until the tops are golden brown and slightly crispy, about 25 minutes. Cool.
Loosen the edges of the tart from the mold using a sharp paring knife. Gently remove the tart mold before plating. Serve with the Cinnamon Mascarpone and Apple Cider Caramel Sauce.
Mix the mascarpone, sugar and cinnamon in a bowl until well incorporated. Refrigerate until ready to use.
Cook the apple cider and lemon juice in a large heavy-bottomed saute pan over medium heat until reduced to a syrupy consistency, about 10 minutes. Stir in the heavy cream and bring back to a boil. Reduce the heat to low and whisk in the butter, 2 pats at a time, allowing the butter to melt before adding more. Serve warm.