Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a 9x13-inch baking dish.

Whisk 2 1/2 cups flour, 2 tablespoons white sugar, and 1/2 teaspoon salt together in a bowl until well blended; cut in shortening until mixture resembles coarse crumbs. Whisk 1/2 cup milk and egg yolks together in a bowl until well mixed; stir into flour mixture until dough comes together.

Divide dough in half. Roll out 1 dough half on a lightly floured surface to a 9x13-inch rectangle; press into the prepared baking dish to form bottom crust. Lightly brush crust with egg whites. Arrange sliced apples on bottom crust; sprinkle with 3/4 cup white sugar and cinnamon. Roll out remaining dough half to a 9x13-inch rectangle; place over apples. Lightly brush top crust with egg whites.

Bake in the preheated oven until crust is golden, 35 to 40 minutes. Cool completely.

Beat confectioners’ sugar, 3 tablespoons milk, vanilla extract, and 1 pinch salt in a bowl until smooth; drizzle over cooled bars, then cut into 20 squares.