In a large bowl, whisk together the flour, sugar, and salt until well combined. Add the cold cubed butter and use a pastry cutter or your fingertips to work it into the flour until the mixture resembles coarse crumbs with some pea-sized butter pieces remaining. Gradually add the ice water, one tablespoon at a time, stirring gently with a fork until the dough begins to come together.
Turn the dough onto a lightly floured surface and shape it into a disk about 1-inch thick. Don't overwork the dough. Wrap the disk in plastic wrap and refrigerate for at least 30 minutes.
While the dough is chilling, core and slice the apples into ¼-inch thick slices. Toss the apple slices with lemon juice in a large bowl to prevent browning.
Add the brown sugar, cinnamon, nutmeg, cloves, cornstarch, and vanilla extract to the apples, then gently mix until the apples are evenly coated with the spice mixture.
In a medium saucepan, heat the sugar over medium heat. Avoid stirring until the sugar begins to melt, then gently swirl the pan occasionally until all sugar has melted and turned amber (about 5-6 minutes).
Immediately add the butter (be careful as it will bubble vigorously) and whisk until incorporated. Remove from heat, count to three, then slowly add the heavy cream while continuing to whisk. Stir in the sea salt. Allow the caramel to cool slightly.
Preheat your oven to 375°F (190°C) and line a baking sheet with parchment paper. On a lightly floured surface, roll the chilled dough into a 12-inch circle about ⅛-inch thick.
Transfer the dough to the prepared baking sheet. Arrange the apple slices in an overlapping pattern in the center of the dough, leaving a 2-inch border all around.
Drizzle half of the caramel sauce over the apples, reserving the rest for serving. Gently fold the edges of the dough over the apples, pleating as you go around the circle.
Beat the egg with 1 tablespoon of water and brush this egg wash over the exposed dough. Sprinkle with coarse sugar if desired.
Bake for 35-40 minutes, or until the crust is golden brown and the apples are tender when pierced with a fork.
Allow to cool slightly before serving. Drizzle with the remaining caramel sauce just before serving.