Place apricots with enough water to cover in medium saucepan. Bring to a boil, recude heat and simmer for 30 mins. Drain and mash. Stir in sugar, lemon juice, cinnamon and nutmeg. Set aside.
Make pastry while apricots are cooking. Combine flour and salt in large mixing bowl; cut in shortening with pastry blender until mixture resembles course meal. Combine egg and water; sprinkle over flour mixture. Add vinegar and lightly stir with a fork until mixture forms a ball. Wrap pastry in wax paper; chill at least 1 hr, or until ready to use.
Roll dough to 1/8" thickness on a lightly floured surface. Cut into circles (4-6"). Place 1 or 2 TB apricot mixture in center of each pastry circle. Moisten edges of pastry, fold in half, making sure edges are even. Press edges of pastry together using a fork dipped in flour. Heat 1" of oil to 375F in a large skillet. Cook pies until golden brown on both sides, turning once. Drain well on paper towels. Sprinkle with powdered sugar, granulated sugar or sugar/cinnamon mixture.
Other fruits may be substituted. Prepare them as you would for a pie.