Place peaches with water to cover in a large saucepan. Bring to a boil; reduce heat and simmer 1 hour. Drain peaches; reserve 1/4 cup liquid. Mash peaches.
Combine peaches, reserved liquid, sugar, lemon juice, cinnamon, and nutmeg; stir well, and set aside.
Divide pastry into 24 portions. Roll each portion into a 3- inch circle on a lightly floured surface.
Place 1 tablespoon peach mixture in center of each pastry circle. Moisten edges of circles; fold pastry in half, making sure edges are even. Using a fork dipped in flour, press edges of pastry together to seal. Prick pastry 2 to 3 times.
Heat 1 inch of oil to 375°. Cook pies until golden brown, turning once. Drain well.