Roll half of pastry to 1/8-inch thickness; line a 9-inch pie plate.
Combine apples, 2/3 cup sugar, brown sugar, orange juice, and water in a medium saucepan; bring to a boil. Reduce heat; cover, and simmer 5 minutes or until apples are tender.
Drain apples, and set aside. Reserve syrup in saucepan; cook over medium heat until volume is reduced by half. Add spices, stirring well.
Spoon apples into pastry-lined pie plate. Pour syrup (about 1/2 cup) over apples, and dot with butter.
Roll out remaining pastry to 1/8-inch thickness, and cut into fourteen 10- x 1/2-inch strips. Arrange strips in lattice design over pie; press ends gently to seal strips to pastry overhang. Brush with egg white, and sprinkle with remaining sugar. Bake at 450° for 10 minutes; reduce heat to 350°; bake 30 minutes or until lightly browned.