Make the dough: In a large bowl, whisk flour and salt. Cut in cold butter with a pastry cutter or fingers until mixture resembles coarse crumbs. Add ice water 1 tbsp at a time, mixing with a fork, until dough just holds together.
Chill dough: Divide dough in half, shape into discs, wrap in plastic, and refrigerate for at least 1 hour.
Make filling: While dough chills, combine diced peaches, sugar, cornstarch, vanilla, cinnamon, and nutmeg in a bowl. Stir well and set aside.
Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper.
Roll and cut: On a floured surface, roll one dough disc to ⅛-inch thickness. Cut into 8 rounds using a 4-5 inch cutter. Repeat with second disc.
Assemble: Place 1 ½ tbsp of peach filling in the center of half the dough rounds. Brush edges with water, top with another dough round, and press edges to seal. Crimp firmly with a fork.
Finish: Place pies on baking sheet. Brush tops with beaten egg and sprinkle with coarse sugar. Cut 2-3 small slits in each top.
Bake for 22-28 minutes, until deep golden brown. Cool on the sheet for 10 minutes before transferring to a wire rack.