Whisk together the flour, granulated sugar, and salt in a large mixing bowl.
Add the cold cubed butter and shortening, then cut them into the flour using a pastry blender or two forks until the mixture looks like coarse crumbs with some pea-size pieces of fat still visible.
Add the ice water one tablespoon at a time, stirring after each addition, until the dough just comes together and no dry flour remains.
Turn the dough out onto a floured surface, divide it in half, and shape each half into a flat disc. Wrap each disc tightly in plastic wrap and refrigerate for at least 30 minutes.
Combine the diced peaches, brown sugar, granulated sugar, lemon juice, cornstarch, cinnamon, nutmeg, and salt in a medium saucepan.
Set the pan over medium heat and cook, stirring often, until the peaches soften and release their juices, about 5 minutes.
Continue cooking, stirring more frequently, until the mixture thickens to a jam-like consistency, about 3 to 5 minutes longer.
Remove the pan from the heat and stir in the 2 tablespoons of butter until melted and glossy. Transfer the filling to a bowl and let it cool completely at room temperature.
Remove one dough disc from the refrigerator and lightly flour a clean work surface.
Roll the dough out to about ⅛ inch thick, rotating it occasionally so it doesn't stick.
Cut circles from the dough using a 4 to 5-inch round cutter, gathering and re-rolling scraps as needed. Repeat with the second disc until you have 10 circles total.
Place about 1 ½ tablespoons of cooled peach filling in the center of each dough circle, leaving a ½-inch border clear around the edge.
Lightly moisten the edge of the circle with a wet fingertip, then fold the dough over the filling to form a half-moon shape.
Press the edges together firmly with your fingers to remove air pockets, then crimp the edge all the way around with the tines of a fork. Set each finished pie on the prepared baking sheet.
Pour the oil into a Dutch oven or other deep, heavy-bottomed pot until it reaches about 2 inches deep.
Clip a deep-fry thermometer to the side of the pot and heat the oil over medium heat to 350°F. Adjust the burner as needed to hold that temperature steady before adding any pies.
Carefully lower 2 to 3 pies into the hot oil at a time, without crowding the pot.
Fry for 3 to 4 minutes, flipping once halfway through, until both sides are deep golden brown.
Use a spider strainer or slotted spoon to transfer the fried pies to the prepared wire rack. Let the oil return to 350°F between batches before adding the next set of pies.
Whisk together the powdered sugar, milk, vanilla extract, and optional cinnamon in a small bowl until smooth and pourable.
Brush or drizzle the glaze over each pie while it's still warm. Let the glaze set for about 5 minutes before serving.