Peel, pit, and cut peaches into 1/2-inch-thick slices. You should have 7 to 8 cups fruit.
In a large bowl, gently mix peaches with cherries, sugar, tapioca, lemon juice, nutmeg, and almond extract.
Unfold 1 pastry round and ease evenly into a 9-inch pie pan. Fill pastry with fruit mixture.
Unfold remaining pastry round on a lightly floured board. Roll pastry evenly to make a 13-inch round.
Center pastry over fruit. Fold top edges of pastry over rim of bottom pastry; flute edges and slash top decoratively. Set pie on a foil-lined baking sheet.
Bake in a 375° oven until juices bubble near center, 1 to 1 1/4 hours. If pastry edges get too dark, lightly cover with foil. Filling firms as pie cools; let stand at least 2 or up to 8 hours before cutting into wedges.