1. Make the pastry: In a large food processor (14-cup or larger), combine the flour, cornstarch, and salt. Pulse to mix. Add the cold cream cheese and pulse until broken down. Add the cold butter and pulse until pea-sized pieces remain. Stream in the ice water mixed with apple cider vinegar while the motor runs. Process just until the dough comes together. (Note: If your food processor is smaller than 14 cups, make the pie dough in two separate batches using half the ingredients for each).
2. Chill the dough: Divide the dough into two portions. Bring them together, wrap tightly in plastic wrap, and roll the wrapped dough out slightly. Refrigerate for at least 2 hours.
3. Prepare the filling: In a large bowl, combine the cherries, strawberries, sugar, salt, cornstarch, lemon juice, and almond extract. Let the mixture sit until the sugar dissolves and the juices release.
4. Assemble the pie: Preheat the oven to 425°F (220°C). Roll out the first dough portion into a 13×17-inch rectangle and press it into an 11×15-inch jelly roll pan. Add the fruit filling in an even layer. Roll out the second portion into a 12×16-inch rectangle. Cut it into strips for a lattice top or leave it whole and cut steam vents.
5. Bake to perfection: Brush the crust with egg wash and sprinkle generously with coarse sugar. Place the jelly roll pan on a larger rimmed baking sheet to catch any drips. Bake on the lower oven rack for 45 to 50 minutes until the crust is golden and the filling is bubbling throughout.
6. Cool and serve: Let the pie cool completely before slicing to allow the filling to set properly.