Preheat the oven to 400 F and place a rack in the lower third. Line a rimmed baking sheet with parchment paper or a silicone baking mat.
Generously flour a work surface and place the chilled dough disk on the floor. Dust the top of the dough with flour. Using a rolling pin, roll chilled dough into a circle, turning and flipping the dough as needed with a bench scraper to prevent sticking until the circle measures 14 inches wide and 1/8-inch thick. Starting at one edge, roll the dough around the rolling pin and transfer to baking sheet. Chill crust while preparing the rest of the ingredients.
Peel, core, and slice apples in 1/8-inch thick pieces. Transfer apples to a large bowl, and add 1/4 cup sugar, brown sugar, flour, spices, and salt. Toss evenly to coat.
Remove the crust from the fridge. Top with the toasted pecans.
Mound the apple filling in the center of the chilled pastry crust, leaving a 3-inch border, and dot filling with the butter. Fold the border up and around the fruit. Working around the border, pinch the dough together to make an attractive pleat. Brush the top of the pleated dough, with the beaten egg and sprinkle border with walnuts and remaining tablespoon of sugar.
Bake crostata until is golden brown and bubbly, 40 to 45 minutes. Use a metal spatula to lift the crostata slightly and check the underside of the crust, which should be golden brown like the top. Transfer the crostata to a rack to cool for 15 minutes. Slice and serve warm topped with vanilla icecream and caramel sauce.