Make Pie Dough: In a large bowl, combine flour and salt. Using a pastry cutter or your hands, cut in the cold butter and shortening until only pea-sized pieces remain. Add 3 tablespoons water and work with your hands to combine. Add up to 2 more tablespoons, 1 tablespoon at a time, until dough clumps. Gently knead dough until it comes together. Divide dough in half, shape each half into a disk, and wrap both disks in plastic wrap. Chill at least 2 hours.

Spray a 12-cup muffin pan generously with cooking spray and set aside. Remove one dough disk from refrigerator. Unwrap and on a lightly floured surface, roll disk out to 1/8-inch thickness. Use a 3 1/2-inch round pastry cutter to cut 12 circles from the dough, re-rolling scraps as needed. Place each circle inside 1 of the muffin cups and use your fingers to press the dough into the cup so that it is even with the bottom and sides of each muffin cup. Once all cups are filled, freeze pan for at least 2 hours.

Preheat oven to 350°F. Transfer muffin pan directly from freezer to oven until the pie dough is almost cooked through, about 13 minutes. Remove from oven and use the back of a spoon to press down any areas that have puffed up during baking. Set aside to cool.

Make Filling: combine apples, sugar, butter, lemon, cinnamon, nutmeg, and salt in a medium saucepan, and stir over medium-high heat until apples have softened and liquid has slightly thickened, about 15 minutes. Remove 2 tablespoons of the hot liquid to a small bowl. Add cornstarch to the bowl and whisk with a fork until no lumps remain. Transfer cornstarch slurry to saucepan with apples and stir in. Bring mixture to a boil for 1 minute before removing from heat.

Scoop 1 tablespoon of apple filling into each muffin cup. Roll out remaining disk of pie dough on a lightly floured surface and using a 2 1/2-inch round pastry cutter, cut out 12 more rounds of dough. Either place these directly on top of the apple filling and press the sides into the bottom crust, or cut each circle into 1/4-inch wide strips and make a lattice on top of each mini pie. Make Egg Wash by whisking together egg and water. Brush the surface of each pie with egg wash. Bake until tops are golden brown, 25 to 30 minutes. Remove pies from muffin pan and allow to cool before serving.