Cook the peach filling: In a medium saucepan, combine peaches, sugars, butter, cornstarch, cinnamon, nutmeg, lemon juice, vanilla, and salt. Cook over medium heat for 8-10 minutes until thickened. Cool completely for 20-25 minutes.
Make the pie dough: Whisk together flour, sugar, baking powder, and salt. Cut in cold butter until coarse crumbs form. Add ice water gradually, stirring until dough clumps together. Knead gently 3-4 times to form a smooth ball.
Chill the dough: Divide dough into 8 equal portions, shape into discs, wrap in plastic, and refrigerate for at least 30 minutes.
Roll and cut: On a floured surface, roll each disc into a 1/8-inch thick circle about 5-6 inches across. Cut into neat rounds using a cutter or bowl.
Fill and seal: Spoon 2-3 tablespoons of cooled filling onto one half of each circle, leaving a 1/2-inch border. Fold over, press edges firmly, and crimp with a fork. Poke 1-2 vent holes in the top.
Fry: Heat oil to 350°F in a cast iron skillet. Fry 2-3 pies at a time for 3-4 minutes, flipping once, until deep golden brown. Drain on paper towels.
Glaze and cool: Whisk powdered sugar, milk, and vanilla until smooth. Brush over warm pies. Let glaze set for 5 minutes. Cool for at least 10 minutes before serving.