Whisk the flour, sugar, and salt in a large bowl. Add the cold butter and cut it in with a pastry blender until most pieces resemble peas, with a few flat almond-sized pieces remaining. Drizzle in three-quarters of the ice water while tossing with a fork, then add the remainder only until the dough holds when squeezed. Divide into two disks, making one slightly larger for the base. Wrap and refrigerate for at least 2 hours.
On a lightly floured surface, roll the larger dough disk into a 13-inch circle about 1/8 inch thick, rotating it frequently to prevent sticking. Transfer it to a 9-inch metal pie pan without stretching, then press it gently into the corners and leave a 1-inch overhang. Roll the second disk into a 12-inch circle, place it on a parchment-lined baking sheet, and refrigerate both crusts while preparing the caramel and filling.
Combine the granulated sugar and water in a medium heavy-bottomed saucepan. Cook over medium heat, without stirring after it begins to boil, until the syrup turns a deep amber color, about 8 to 12 minutes. Remove the pan from the heat and gradually whisk in the warm cream, protecting your hands from the vigorous steam. Whisk in the butter, salt, and vanilla until smooth. Measure 3/4 cup for the filling and reserve the remaining caramel for serving.
Combine the sliced apples, brown sugar, cornstarch, cinnamon, nutmeg, salt, and lemon juice in a large Dutch oven. Cook over medium heat for 10 to 14 minutes, folding frequently with a silicone spatula so the slices heat evenly without breaking. Stop when the apples bend slightly, the released liquid bubbles, and the sauce looks thick and translucent. Remove from the heat, fold in 3/4 cup caramel, spread the filling on a rimmed sheet, and cool for 25 minutes.
Spoon the cooled apple filling into the lined pie pan, arranging the slices compactly and mounding them slightly in the center. Lay the second pastry round over the fruit, trim both layers to a 1-inch overhang, and fold the edges underneath. Crimp firmly, then cut five vents in the top. Beat the egg with one tablespoon water, brush a thin layer over the pastry, and sprinkle with turbinado sugar. Refrigerate the assembled pie for 20 minutes.
While the assembled pie chills, position a rack in the lower third of the oven, place a rimmed metal baking sheet on it, and heat the oven to 425°F for at least 20 minutes. Set the pie on the hot sheet and bake for 20 minutes. Reduce the temperature to 375°F and continue baking for 45 to 55 minutes, shielding the edge if it darkens too quickly. The crust should be deep golden and thick filling should bubble through the vents.
Transfer the pie pan to a wire rack and cool for at least 4 hours before cutting. This rest is part of the setting process, not decorative patience, because slicing while warm releases the thickened juices. If the reserved caramel has become firm, warm it gently in short intervals until pourable but not hot. Cut the cooled pie with a sharp knife, wipe the blade between servings, and drizzle each slice with a portion of the reserved caramel.
