CRUST: In a food processor, pulse 2 1/2 cups flour and the salt. Add butter, and pulse until mixture resembles coarse meal. Drizzle ice cold water evenly over the mixture. Pulse until mixture just begins to hold together. Add cheese; pulse until combined.
Turn out dough onto a surface and form into a ball. Divide the ball in half. Flatten into 2 disks, wrap each disk in plastic and refrigerate until chilled, about 30 minutes or up to overnight. (You can also freeze your other disk of dough for up to one month.)
On a lightly floured surface, roll out 1 disk of dough into a 13-inch round, about 1/8 inch thick. Transfer to a parchment-lined baking sheet.
FILLING: In a large bowl, toss together apples, granulated sugar, 2 tablespoons flour, lemon juice, and cinnamon.
Spread filling over dough, leaving a 2-inch border. Fold edges in, to form a crust; brush with egg wash, and sprinkle with turbinado sugar.
Preheat oven to 375 degrees. Bake the crostata until the apples are tender and the crust is golden, about 1 hour. (Cover the crostata with aluminum foil about 40 minutes into the baking, to prevent any burning.) Let cool slightly.
Brush the apples with warm apricot preserves. Let the crostata cool to room temperature before serving.