Whisk the flour, sugar, and salt together. Cut in the cold cubed butter until coarse, pea-sized pieces remain, then add ice water gradually just until the dough holds together. Divide into two disks, wrap, and refrigerate for at least 30 minutes.

Combine the diced apples, brown sugar, flour or cornstarch, cinnamon, nutmeg, lemon juice, and salt in a saucepan. Cook over medium heat for 5–8 minutes, stirring often, until the apples are slightly softened and the juices have thickened. Cool the filling completely.

Roll the chilled dough to about 1/8-inch thickness. Cut into 4- to 5-inch rounds or rectangles, rerolling scraps only once or twice so the pastry stays tender.

Place approximately 1 1/2–2 tablespoons of cooled apple filling in the center of each pastry piece. Brush the edges lightly with water, fold the dough over the filling, press to seal, and crimp the edges with a fork.

Cut two or three small slits in each pie for ventilation. Arrange the pies on a parchment-lined baking sheet and chill for 15–20 minutes.

Beat the egg with 1 tablespoon water and brush it over the chilled pies. Sprinkle with granulated sugar if desired, then bake at 375°F (190°C) for 20–25 minutes, until deeply golden brown.

Cool the pies for at least 20 minutes. Whisk the confectioners’ sugar, milk, vanilla, and optional salt until smooth, then drizzle or spread the glaze over the cooled pastries and let it set before serving.