Make pastry: Place flour, salt, butter, and shortening in food processor and process just until mixture resembles coarse crumbs. Or mix flour and salt in mixing bowl and cut int butter and shortening with pastry blender. Sprinkle about 5 T. water over flour mixture; process or mix with fork until dough comes together, adding remaining water as needed. Divide dough into 2 pieces, one twice as big as the other. Refrigerate both pieces wrapped in plastic wrap at least 30 minutes.
Meanwhile make filling: Mix sugar and cornstarch in large bowl. Add cherries, vanilla and almond extracts, and pinch salt; stir to combine. Let stand, stirring occasionally, 20 minutes.
Heat oven to 425 degrees.
Roll out larger piece of pastry to 14" circle on well-floured surface. Line 9" deep dish pie plate with pastry and trim to a 1" overhang. Mix egg and 1/4 t. salt and brush over bottom and side of pastry. Stir cherries and pour into pastry. Top with butter.
Roll out remaining dough to 10" circle. Cut into 1/2" strips. Arrange strips in lattice pattern over filling, painting ends with egg glaze to stick to bottom crust. Fold over edges and crisp. Brush rim and lattice with glaze.
Bake 15 minutes. Reduce heat to 375 degrees and bake until crust is golden and cherries are tender, 30-40 minutes longer. If edges brown too quickly, cover with strips of aluminum foil.