Preheat the oven to 350 degrees F.
In a large saute pan, melt the butter. Add the apples and saute for 2 minutes. Add the sugars, lemon juice and flour. Continue to saute for 2 minutes. Season the apples with nutmeg, cinnamon, and salt. Mix thoroughly and remove from the heat. Mix in the pecans. Cool the mixture.
Lightly dust a surface with flour. Cut the dough in half. Roll out each half to 12 inches in diameter and about 1/8-inch thick. Fold 1 circle of dough in fourths. Carefully lift the dough and place in a 10-inch deep-dish pie pan. Unfold the pie dough and spoon the apples into the pie shell. Place the second round of dough over the apples. Using a sharp knife cut away the excess dough. Using your fingers, crimp the edges of the pie firmly to seal the dough completely. With the same sharp knife, make 3 slits, about 4 inches long and 2 inches apart, across the pie dough. Brush with the beaten egg, place the pie in the oven, and bake for 1 hour and 15 minutes.
Remove the pie from the oven and sprinkle the cheese over the top. Return the pie to the oven and continue to cook for 8 minutes or until the cheese is bubbly. Let cool to room temperature before slicing, about 1 hour. Serve with vanilla ice cream.
Sift the flour, sugar, and salt into a large bowl. Using your fingers, work in the butter and shortening until the mixture resembles coarse crumbs. Add 4 tablespoons of ice water and work with your fingers until the water is incorporated and the dough comes together. Add more water as needed just until the dough comes together, being careful not to over mix. Form the dough into a disk, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes before using.