Pie dough:

1. Place the flour, butter, shortening and salt into a stand mixer bowl. Mix on low speed until the fats are in small chunks and the mixture looks a bit dry (about 10-15 seconds). Do not overmix it.
2. Add the ice water all at once and mix on medium speed until the dough just comes together. Some small lumps of fat should remain in the dough.
3. Shape the dough into 3 balls about the same weight (I use a scale). Wrap each one in plastic wrap and refrigerate for at least 30 minutes. At this point you can freeze 1 ball of dough that you are not going to use.

Assembly:

1. Remove the dough from the fridge 10 minutes before you start rolling. Lightly flour a rolling pin and the work surface.
2. Pressing the rolling pin on the dough, slightly flatten it, then start rolling from the center outward. After each stroke of the rolling pin, rotate the dough a quarter turn clockwise and roll again, lightly dusting more flour on the work surface, or on the dough, as needed, to prevent the dough from sticking.
3. Continue rolling until the dough is -more or less- a 12-inch circle and about 1/4-inch thickness.
4. Carefully fold the dough in half and lay it across one side of a buttered pie pan, placing the seam of the dough in the center of the pan. Gently unfold the dough, easing the dough without stretching it, down into the pan, and up the sides. Trim the pastry 1/2 inch beyond pie plate, crimping edges as desired. Cover with plastic wrap and refrigerate for 30 minutes to 1 hour.
5. In the meantime, prepare the filling. Combine the cherries, berries, cornstarch, sugar, salt, vanilla, almond extract, and lemon juice in a large saucepan over medium-high heat. Simmer, stirring almost constantly, until the mixture starts to thicken, about 8-10 minutes. Remove from the heat, transfer the mixture to a heatproof bowl and cool to room temperature.
6. Preheat oven to 400° F (200°C).
7. Roll the remaining dough into a 12-inch circle. Cut pastry with a fluted pasta cutter (or a pizza roller) into strips 1 1/2-inch wide. Weave strips over the fruit mixture in a lattice pattern. Cut excess length and press the ends of the strips into bottom pastry rim. You might have some leftover pastry dough.
8. In a small bowl, whisk together the egg and water. Brush pastry with egg wash and sprinkle with turbinado or cane sugar.
9. Place the pie on a baking sheet and bake for 10 minutes at 400° F, then reduce the oven temperature to 375° F and continue baking for 55-60 minutes, or until the crust is golden and the filling is bubbling. Cover with foil, if necessary, to prevent excess browning.
10. Remove from the oven, place pie plate on a rack and let cool for at least 2-3 hours before slicing.