In large mixing bowl, stir together the 5 cups flour, the 1/4 cup granulated sugar, and 1 teaspoon salt. Cut in shortening until pieces are pea-size.
In a small bowl, stir together egg yolks and 1 cup milk. Add to flour mixture; stir until flour mixture is just moistened. Form dough into two balls.
On a lightly floured surface, roll one ball of dough into a 17x13-inch rectangle. Ease pastry into a 15x10x1-inch baking pan (should be a 1/2-inch pastry border around the pan).
Arrange apple slices evenly over pastry in pan. In a small bowl, stir together the 1-1/2 cups granulated sugar, the 1/4 cup flour, cinnamon, and 1/4 teaspoon salt. Sprinkle evenly over apples. Dot with butter.
Roll out remaining dough into a 16x11-inch rectangle. Place pastry over top of filling. Fold over bottom crust. Crimp edges together. Brush top of pastry with the beaten egg white. Bake in a 350 degree F oven for about 45 minutes or until the pastry is golden and the apples are tender. Cool slightly.
In a medium bowl, stir together powdered sugar, vanilla, and enough milk to make a thick frosting. Spread over top of pastry. Cool. Cut into squares. Makes 24 squares.