Instructions

1. Make the pastry crust: In a large bowl, whisk the flour with the sugar and salt. Using your fingers, rub the lard and butter into the flour until they form pea-size crumbles. Add the vodka and water and stir until the dough just comes together. Divide the dough in half and flatten each half into a disk. Wrap each disk in plastic wrap and refrigerate for at least 1 hour before using.
2. Meanwhile, make the caramel: In a small skillet, combine the cream with the butter and rosemary. Heat over high and cook for 3 minutes. Remove the cream from the heat and keep warm. In a small saucepan, combine the sugar and corn syrup and heat over high. Cook the sugar, without stirring, until golden, 7 to 8 minutes. Remove the pan from the heat and pour the warm cream into the caramel, and stir until the caramel dissolves. Let the caramel stand at room temperature for 2 hours to infuse flavors, and then remove the rosemary.
3. Heat the oven to 375°. On a lightly floured work surface, roll 1 dough disk into a 12-inch circle. Fit the circle into a 9-inch deep-dish pie plate, trim the edge with a knife, leaving a 1-inch overhang, and brush the edge with some of the beaten egg. In a large bowl, combine 1 cup of the rosemary caramel with the apples, cornstarch, and salt and scrape the filling into the pie crust. Roll the remaining dough into a 12-inch circle and place over the apples. Trim the excess dough, and fold the overhang under to form a thick edge. Crimp the edge with your fingers to seal.
4. Brush the top of the pie with the remaining egg and, using a knife, make 4 slits in the top. Transfer the pie to the oven and bake until golden, about 1 hour. Pour 1⁄2 cup of the remaining rosemary caramel over the top of the pie, and continue baking until the caramel is lightly set, about 15 minutes more. Remove the pie from the oven and transfer to a rack. Sprinkle with sea salt and let cool to room temperature. Serve warm slices of pie with ice cream on top, if you like.