Mix apple slices with 1/2 cup granulated sugar, 3 tablespoons cornstarch, 1 tablespoon lemon juice, and ground cinnamon and nutmeg. Taste and, if desired, add more granulated sugar and lemon juice.
On a lightly floured board, roll half the pastry into a round 1/8 inch thick. Line a 9-inch pie pan with pastry. Fill with apple mixture.
On a lightly floured board, roll remaining pastry into a 1/8-inch-thick round and lay over apple mixture. Fold edges of top pastry over edges of the bottom one and crimp to seal together. Cut decorative slits in top pastry and sprinkle with about 1 tablespoon granulated sugar.
Bake on the lowest rack in a 375° oven until juices bubble in center of pie, 1 to 1 1/4 hours. If pastry edges brown before pie is done, drape affected areas with foil. Cool pie on a rack at least 2 hours.
Meanwhile, in a 1 1/2- to 2-quart pan, combine apple juice, cinnamon stick, and orange peel. Cover and simmer over low heat for 15 minutes. Stir in brown sugar until it dissolves. Mix remaining 1 1/2 tablespoons cornstarch smoothly with 3 tablespoons water; stir into juice mixture over high heat until sauce boils. Discard cinnamon stick and orange peel.
Cut warm or cool pie into wedges; top each portion with vanilla ice cream and warm or cool cinnamon sauce.