Prepare the Filling:
Peel, core, and dice the apples.
In a medium bowl, combine diced apples, brown sugar, cinnamon, vanilla extract, apple cider, and cornstarch. Stir to combine and set aside.

Prepare the Dough:
In a large bowl, whisk together the sifted flour and salt.
Cut in the cold, cubed butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.
In a small bowl, whisk together the egg yolk and cold milk.
Gradually add the egg yolk mixture to the flour mixture, mixing until just combined. Be careful not to overmix.
Gather the dough into a ball, flatten into a disc, wrap in plastic wrap, and refrigerate for at least 30 minutes.

Assemble the Pies:
On a lightly floured surface, roll out the chilled dough to about 1/8-inch thickness.
Cut out 6-inch circles using a plate or a large cookie cutter.
Place about 2-3 tablespoons of the apple filling onto one half of each dough circle, leaving a 1/2-inch border.
Moisten the edges of the dough with a little water.
Fold the other half of the dough over the filling to create a half-moon shape.
Crimp the edges firmly with a fork to seal.
Cut a small slit or two in the top of each pie to allow steam to escape.

Fry the Pies:
Heat the vegetable oil in a large, deep skillet or Dutch oven over medium-high heat to 350°F (175°C).
Carefully slide 2-3 pies into the hot oil, ensuring not to overcrowd the pan.
Fry for about 3-4 minutes per side, until golden brown and crisp.
Use a slotted spoon or tongs to remove the pies from the oil and place them on a wire rack set over a baking sheet to drain.
Repeat with the remaining pies.

Prepare the Glaze and Finish:
In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth. Add more milk if needed to reach desired consistency.
While the pies are still warm, drizzle the glaze over the tops.
Let the glaze set slightly before serving.