How to Make It
Make the pastry In a food processor, pulse the 2½ cups of flour with the salt. Add the butter and pulse until pea-size. Add 1/4 cup of ice water and pulse until the dough just comes together. If needed, add up to 1/4 cup of ice water. Turn out onto a work surface and knead 2 or 3 times. Quarter and press into rectangular disks. Wrap in plastic and refrigerate until firm, at least 1 hour.
Make the fillings In the bowl of a stand mixer, combine the pork with the breadcrumbs, sage, garlic, 1 teaspoon of salt, 1/4 teaspoon of pepper, 1/4 teaspoon of granulated sugar, 1/4 teaspoon of ginger, 1/8 teaspoon of allspice and a pinch of nutmeg. Freeze until cold but not frozen, about 30 minutes.
Using the paddle attachment, beat the pork mixture at low speed until just combined. Slowly beat in the stock.
In a medium bowl, toss the apples with the brown sugar, cornstarch, lemon juice, cardamom, cinnamon and the remaining 1/4 teaspoon of ginger and 1/4 teaspoon of allspice.
On a lightly floured work surface, roll out 1 disk of dough to a 5-by-10-inch rectangle. With a long side nearest you, spoon one-fourth of the apples along the bottom right half of the dough, leaving a 1-inch border on the bottom and side. Spoon one-fourth of the pork filling along the bottom left half of the dough to make 1 solid bar of filling, leaving a 1-inch border on the bottom and side. Fold the top half of the dough over the filling and press the edges to seal. Brush the top with the beaten egg. Sprinkle the apple half with granulated sugar and cut a few slits in the top of this side only. Transfer to a parchment-lined baking sheet. Repeat with the remaining dough and filling, spacing the pies 1 inch apart. Refrigerate until chilled, about 30 minutes.
Preheat the oven to 400°. Bake the pies for 50 minutes to 1 hour, rotating the baking sheet halfway through, until the apple filling is bubbling and the crust is deeply golden. Transfer the pies to a rack to cool.