Make the dough. Whisk the flour, salt, and sugar in a large bowl. Cut in the butter with a pastry cutter or your fingertips until the pieces are about the size of peas. Stir in the cheddar. Drizzle in the ice water one tablespoon at a time, tossing with a fork, until the dough just holds together when you squeeze a handful. It should not feel wet. Flatten into a disk, wrap tightly, and refrigerate for 30 minutes.
Cook the filling. Melt the butter in a wide skillet over medium heat. Add the apples, brown sugar, lemon juice, cinnamon, nutmeg, and salt. Cook, stirring often, for 8 to 10 minutes, until the apples are just tender at the edges and the juices run syrupy. Stir the cornstarch into 1 tablespoon of cold water, pour it in, and cook 1 minute more until the filling turns glossy. Scrape into a bowl and let it cool completely. Warm filling melts the crust before it ever reaches the oven.
Roll and cut. Heat the oven to 400 degrees F and line a baking sheet with parchment. Roll the dough on a lightly floured surface to about 1/8 inch thick. Cut eight 5-inch rounds with a bowl or large cutter, gathering and re-rolling the scraps once. If the dough softens, slide it back into the fridge for 10 minutes before you cut.
Fill and seal. Spoon about 3 tablespoons of filling onto one half of each round, leaving a 1/2-inch border. Fold the dough over and press the edges firmly with your fingers, then crimp with the tines of a fork. Cut two small vents in the top of each pie so steam has somewhere to go. Do not overfill them. A pie that is too full is a pie that leaks.
Chill, wash, and sugar. Set the filled pies on the baking sheet and chill for 10 minutes. Brush the tops with the beaten egg and sprinkle with the coarse sugar.
Bake. Bake for 22 to 26 minutes, until the crusts are deep golden and the filling bubbles gently at the vents. Cool on the sheet for 10 minutes before you move them. The filling comes out of the oven hot enough to burn, and it needs that time to set.