On a lightly floured surface, roll 1 disk of dough into a 12″ round. Fit into a 9″ deep-dish pie plate. Trim edges with a knife, leaving 1″ dough overhanging edge of plate; chill 30 minutes. On a lightly floured piece of parchment paper, roll remaining disk of dough into a 12″ round; trim, discarding edges, into a 9″ square. Slice into six 1 1⁄2″-thick strips; chill 30 minutes.
Toss apples with 2 tbsp. granulated sugar and lemon juice; cover with plastic wrap and set aside 1 hour. Whisk 1 cup sugar and 1⁄4 cup water in a 2-qt. saucepan over medium heat; cook, without stirring, until the sugar dissolves, 2–3 minutes. Add butter; bring to a boil. Continue cooking until the mixture turns a deep red-brown, about 25 minutes or until a candy thermometer inserted in sauce reads 325°. Remove from heat. Carefully add cream, stirring until sauce is smooth.
Heat oven to 375°. Stir 1⁄3 cup Demerara sugar with 2 tbsp. flour, plus bitters, nutmeg, allspice, cinnamon, kosher salt, and pepper in a large bowl. Drain apples, discarding juices, and add to bowl; toss to coat in spice mixture. Sprinkle remaining flour and sugar over crust; tightly arrange apples over dough, mounding them slightly higher in the center. Pour caramel sauce evenly over apples. Sprinkle with half the sea salt. Lay three strips of dough vertically across the top of pie, about 1 1⁄2″ apart. Fold first and third strips in half upward. Lay one horizontal strip at pie's center. Unfold first and third vertical strips over it. Fold second vertical strip upward. Lay another horizontal strip below the first. Unfold second vertical strip over the bottom horizontal strip, then fold it again downward. Lay a third horizontal strip just above the pie's center and unfold second vertical strip. Pinch bottom crust and lattice edges together; roll toward center of pie to hide lattice edges; crimp edges. Brush egg over crust, then sprinkle lattice with remaining Demerara sugar and sea salt. Bake until crust is golden and filling is bubbly, about 1 hour. Let cool completely before serving.