Place sliced peaches, sugars, butter, lemon juice, cinnamon, and nutmeg in a medium saucepan. Cook over medium heat for 10-15 minutes, stirring occasionally, until peaches soften and liquid thickens.
Mash peaches with a wooden spoon or potato masher into a chunky puree. If needed, whisk cornstarch with 1 tablespoon cold water and stir into filling; cook 2 minutes more. Cool completely to room temperature.
Whisk flour, baking powder, and salt in a large bowl. Cut in cold shortening with a pastry cutter or fork until mixture resembles coarse crumbs with pea-sized pieces.
Pour in cold milk and stir with a fork just until dough comes together in a shaggy mass. Turn onto a floured surface, shape into a disc, wrap in plastic, and refrigerate for 20-30 minutes.
On a floured surface, divide dough into 8 pieces. Roll each into a ⅛-inch (3 mm) thick circle about 5-6 inches (12-15 cm) in diameter. Use a round cutter or bowl for perfect circles.
Place a dough circle flat. Spoon 2-3 tablespoons cooled filling onto one half, leaving a ½-inch border. Moisten edges with water, fold dough over filling, press out air, and crimp firmly with a fork.
Place assembled pies on a parchment-lined baking sheet and refrigerate for 15-20 minutes.
Heat oil in a heavy skillet to 350-375°F (175-190°C). Carefully lower 3-4 chilled pies into hot oil and fry 3-4 minutes per side until deep golden brown.
Remove pies with a slotted spoon and drain on a wire rack over paper towels. Allow oil to return to temperature between batches.
Whisk powdered sugar, milk, and vanilla in a small bowl until smooth. Drizzle or dip glaze over warm pies. Let set for 5 minutes before serving.