Whisk the flour, sugar, and salt together. Cut in the very cold butter until pea-sized pieces remain, then add ice water one tablespoon at a time until the dough holds together when pressed.

Divide the dough into two disks, wrap each one, and refrigerate for at least 30 minutes. Combine the peaches, sugar, cornstarch, lemon juice, cinnamon, salt, and optional vanilla; cook briefly over medium heat if very juicy, then cool completely.

Roll the chilled dough to about 1/8-inch thickness and cut eight roughly 5-by-3-inch rectangles or 5-inch circles. Place 1 1/2 to 2 tablespoons of filling on half of the pieces, leaving a border.

Brush the edges with water, top with the remaining dough pieces, and crimp with a fork. Cut one or two small vents in each pie, then refrigerate the assembled pies for 15–20 minutes.

Heat the oven to 400°F (200°C). Brush each chilled pie with the beaten egg mixed with water or milk, sprinkle with coarse sugar if using, and bake for 20–25 minutes until deeply golden and bubbling.

Cool the hand pies on the baking sheet for at least 15 minutes. Whisk together the confectioners’ sugar, milk or cream, and vanilla, then drizzle the glaze over the cooled pies if desired.