Process the flour, sugar, and salt in a food processor until blended. Add the butter; pulse for about 5 seconds or until the mixture resembles coarse crumbs. Add the water; pulse for about 5 seconds, just until moist clumps form. Gather the dough into a ball; flatten into a disk. Wrap in plastic wrap and refrigerate for at least 1 hour.
Heat the oven to 375° F.
Mix the brown sugar, flour, and cinnamon in a large bowl; add the apples and toss to coat.
Roll out dough to a 13-inch round between two sheets of parchment paper. Remove and discard top parchment paper. Slide the pastry onto a baking sheet, keeping the bottom paper in place.
Arrange the apple slices in overlapping concentric circles to within 2 inches of the edge of the pastry. Fold the edge of the pastry over the apples using the parchment paper as an aid. Overlap the pastry slightly while folding; press gently to seal. Brush crust with beaten egg; sprinkle with sugar.
Bake for 35 to 40 minutes or until crust is golden brown. Drizzle filling with three tablespoons salted caramel sauce. Cool for at least 15 minutes. Serve warm with additional caramel sauce and whipped cream or vanilla ice cream, if desired.
The dough can be made up to three days in advance and stored in the refrigerator. Let it sit at room temperature for about 15 minutes before rolling out.