Whisk the flour, granulated sugar, and salt together in a large bowl. Cut in the cold cubed butter until coarse crumbs form, keeping some pea-sized butter pieces visible for a flaky pastry.

Add the egg and milk, mixing only until the dough comes together. Divide the dough in half, flatten each portion into a disk, wrap, and refrigerate for at least 30 minutes.

Roll the chilled dough to approximately ⅛-inch thickness. Cut sixteen matching rectangles, roughly 3 by 4 inches, rerolling scraps only as needed.

Place about 1½ tablespoons of thick jam in the center of eight pastry rectangles, leaving a clear border around each one. If the jam is thin, mix it with 1 tablespoon cornstarch before filling.

Whisk the egg wash egg with 1 tablespoon milk or water. Brush the exposed pastry borders, cover each filled piece with a remaining rectangle, press the edges together, and crimp firmly with a fork.

Move the pastries to a parchment-lined baking sheet and refrigerate for 15–20 minutes. Pierce the tops several times with a fork or small skewer to vent steam.

Brush the pastries lightly with egg wash. Bake at 350°F (175°C) for 15–18 minutes, or until the edges are lightly golden.

Cool the pastries completely before decorating. Stir together the confectioners' sugar, milk, and vanilla, tint portions red or green if desired, then spread the icing over the pastries and add decorations.