Make the filling: In a skillet over medium heat, melt butter. Add diced apples, brown sugar, granulated sugar, cinnamon, nutmeg, salt, and lemon juice. Cook 5–7 minutes, stirring, until apples are tender but not mushy.
Thicken: Stir cornstarch with water to make a slurry. Add to the skillet and cook 1–2 minutes until the juices turn glossy and thick. Transfer filling to a bowl and cool completely.
Prep dough: On a lightly floured surface, roll pie dough to about 1/8-inch thickness. Cut into 4–5 inch circles. Gather and gently re-roll scraps once if needed.
Fill: Spoon 1–2 tablespoons cooled apple filling into the center of each circle. Brush edges lightly with egg wash.
Seal: Fold dough over to form a half-moon. Press edges to seal, then crimp with a fork. Check for gaps and patch with a dab of egg wash if needed.
Heat oil: Pour 2 inches of oil into a heavy skillet. Heat to 350°F (175°C). Maintain temperature between 350–360°F for best results.
Fry: Working in small batches, lower pies into hot oil. Fry 2–3 minutes per side until deep golden brown. Avoid crowding, which lowers oil temperature.
Drain: Transfer pies to a cooling rack set over paper towels to drain and stay crisp.
Glaze: Whisk powdered sugar, vanilla, and milk to a pourable consistency. Drizzle over warm (not hot) pies, or dust with powdered sugar.
Serve: Enjoy warm. For baking, brush with egg wash and bake at 375°F (190°C) for 20–25 minutes until golden. For air-frying, 375°F for 10–12 minutes, flipping halfway.