Place apricots with water to cover in a medium saucepan. Bring to a boil; reduce heat, and simmer 30 minutes. Drain well, and mash. Stir in sugar, lemon juice, cinnamon, and nutmeg; set aside.
Roll dough to 1/4-inch thickness on a lightly floured surface. Cut into 4-inch circles.
Place 1 tablespoon apricot mixture in center of each pastry circle. Moisten edges of pastry; fold in half, making sure edges are even. Press pastry edges firmly together using a fork dipped in flour.
Heat 1 inch of oil to 375° in a large skillet. Cook pies until golden brown on both sides, turning once. Drain well on paper towels. Sprinkle with powdered sugar, if desired.
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