Heat 2 large heavy skillets over high heat. Melt 2 tablespoons butter in each. When the butter bubbles, add the apples and sauté about 3 minutes.
Add the sugar, vanilla, 2 teaspoons of the cinnamon, and the ginger. Stir or toss to coat the apples. Cook, stirring or tossing, frequently, until the apples are brown and juicy, 7 to 8 minutes.
Add the apricots, raisins, 1/2 cup of the cream, and the Calvados, plus a pinch of salt. Cook, stirring or tossing, until the apples are caramelized and the juices form a thick sauce, 7 to 10 minutes. Transfer to a baking sheet to cool.
Dust a work surface with flour. Roll out the round piece of dough to a 12-inch-diameter round, dusting lightly if sticking and turning often. Press together any tears. Transfer to a 9-inch deep-dish pie plate.
Dust the work surface with flour again. Roll out the second piece of dough to dimensions greater than the top of the pie and about 1/4 inch thick. Using a pizza wheel or a sharp knife, cut the square into even 1/2-inch-wide strips.
Position a rack in the center of the oven and preheat to 350°F. Transfer the cooled filling to the dough-lined pie dish. Press the filling slightly to compact. Trim the bottom dough to overhang about 1/2 inch around the edge.
Arrange 6 or 7 dough strips evenly across the filling. Arrange 6 or 7 strips at right angles to the first ones. Trim the ends. Fold the overhang over the lattice ends and crimp together into a neat border, rolling the rim toward the filling.
In a small bowl, stir together the remaining 1 teaspoon ground cinnamon with the granulated sugar. Brush the lattice strips with beaten egg white. Sprinkle the cinnamon sugar over the strips.
Place the pie on a large rimmed baking sheet. Bake until the crust is deep brown and the filling is bubbling, 45 minutes to 1 hour. Leave the pie at room temperature to cool completely, at least 2 hours.
Whip the remaining 1 cup cream to soft peaks. Cut the pie into wedges and transfer to serving plates. Drizzle the Caramel Sauce over and around each slice and add dollops of whipped cream.