Combine cranberries, sugar, water, and tapioca in a medium saucepan; let stand 5 minutes. Bring to a boil; reduce heat. Cook mixture over medium heat, stirring constantly, until thickened. Stir in butter. Set aside to cool.
Divide pastry into thirds; roll each portion to 1/8-inch thickness on waxed paper. Cut into 3 1/2-inch circles.
Place 1 teaspoon cranberry 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 2 to 3 inches of oil to 375° in a large skillet. Cook pies 4 to 5 at a time until golden brown on both sides, turning once. Drain well on paper towels. Sprinkle with powdered sugar.