To make the dough, in a food processor, pulse the flour, salt, and granulated sugar together until combined, about 5 pulses. Add the butter and process until the mixture resembles coarse meal, about 10 pulses. Add 3 Tbs. of the ice water and pulse 2 or 3 times. The dough should hold together when squeezed with your fingers but should not be sticky. If it is crumbly, add more water 1 tsp. at a time, pulsing twice after each addition. Turn the dough out onto a work surface, divide in half and shape each half into a disk. Wrap the disks separately in plastic wrap and refrigerate for at least 2 hours or up to overnight.
On a lightly floured work surface, roll out half of the dough into a 12-inch round about 1/8 inch thick. Fold the dough in half and then into quarters and transfer it to a 9-inch deep-dish pie dish. Unfold and gently press the dough into the bottom and sides of the dish. Trim the edges flush with the rim of the dish. Refrigerate for 30 minutes.
On a large sheet of lightly floured parchment paper, roll out the remaining dough disk into a 12-inch round about 1/8 inch thick. Using leaf pie cutters, make rows of cutouts in the dough, spacing them 1/2 inch apart and leaving a 1-inch border along the edges, to create a lattice appearance. Reserve the cutouts for decorating. Reroll the dough scraps to make more cutouts. Refrigerate the lattice top and cutouts for 30 minutes.
Meanwhile, make the filling: In a large Dutch oven, stir together the apples, brown sugar, granulated sugar, cinnamon, salt, nutmeg and cornstarch. Set over medium heat, cover and cook, stirring occasionally, until the apples are just tender but not mushy, about 20 minutes. Uncover and cook until the liquid has thickened and become glossy, 5 to 7 minutes more. Remove from the heat, stir in the lemon juice and let cool to room temperature, about 30 minutes.
Position a rack in the lower third of an oven, place a baking sheet on the rack and preheat the oven to 400°F.
Let the pie shell, lattice top and leaf cutouts stand at room temperature for 5 minutes. Transfer the apple filling to the pie shell, scatter the butter pieces on top, and gently invert the lattice top over the pie. Trim the edges flush with the rim of the dish and press the top and bottom crusts together to seal. Brush the underside of the leaf cutouts with egg wash and gently press them onto the edges of the piecrust, overlapping the leaves slightly. Decorate as desired with the remaining leaves. Brush the entire top crust with egg wash and sprinkle with the granulated sugar.
Place the pie dish on the preheated baking sheet. Bake until the crust is crisp and golden brown, about 1 hour, covering the edges with aluminum foil if they become too dark. Transfer the pie to a wire rack and let cool for at least 1 1/2 hours before serving. Serves 8.