Directions:

1. Make caramel-apple filling: Heat a large Dutch oven over high heat for 5 minutes. Meanwhile, in a large bowl, toss apples with vanilla seeds and pods, sugar, spices and lemon zest. Transfer apple mixture to pot and sauté until apples start to soften, about 10 minutes. Meanwhile, in a small bowl, combine cornstarch with lemon juice. Set aside. Pour caramel sauce over apples and simmer, stirring occasionally, 5-10 minutes more. Pour cornstarch mixture into caramel-apple mixture and bring to a boil, stirring often, until apples are tender but still hold their shape, 5 minutes. Transfer apple filling to a shallow baking dish and allow to cool completely. (Filling can be made in advance and kept in the refrigerator up to 3 days.)

2. Make crust: In a stand mixer with a paddle attachment, beat flour, sugar, salt and butter until texture resembles sand. Decrease speed to low. Slowly add egg yolks and ice water to flour mixture, mixing until dough just starts to form, about 3 minutes. If dough does not come together, add more ice water, 1 tablespoon at a time, until dough forms. Shape dough into two discs, one slightly larger than the other, wrap in plastic and refrigerate at least 30 minutes or up to a day.

3. Grease an 8-inch diameter, 3-inch-high springform pan with butter. On a floured surface, roll larger dough disk into a circle ⅛-inch thick and 15 inches across. Transfer dough to pan, pressing it into bottom and sides. Refrigerate 20 minutes.

4. On a floured surface, roll second dough disk into a circle ⅛-inch thick and 15 inches across. Use a pizza cutter to cut dough into fourteen ¾-inch-widestrips. Place strips on a baking sheet and refrigerate 20 minutes.

5. Remove springform pan from refrigerator and fill crust to rim with apple filling. Trim excess dough.

6. Make lattice weave: Remove dough strips from refrigerator. Place seven strips horizontally across top of pie, using longer strips in center and shorter strips at ends. Leave a ¼-inch gap between strips. Starting with top strip, fold every other strip back onto itself. Take longest remaining strip and lay it vertically across center so it is perpendicular to horizontal strips. Place folded strips back over vertical strip. Starting with second horizontal strip from top, fold every other strip back onto itself. Place a new strip to right of first vertical strip. Place folded strips over new strip, as before. Repeat pattern with two more strips until a tight lattice covers half of pie. Using remaining three strips, repeat lattice over left half of pie. Trim any excess dough hanging off the sides. Use your fingers to crimp edges together tightly, making sure top crust is well sealed onto bottom crust.

7. Make egg wash: In a small bowl, whisk together egg, water and salt. Brush top crust with egg wash and sprinkle generously with Demerara sugar. Freeze pie 30 minutes to set.

8. Place rimmed baking sheet in oven. Preheat oven to 400 degrees. Place pie on rimmed baking sheet and bake until crust is light golden, about 30 minutes. Reduce temperature to 350 degrees and bake until the filling is bubbling, 30 minutes more. Remove from oven and let cool completely before serving. Pie can be stored at room temperature or refrigerated for two days.