If you don't have a cherry pitter, try inserting a wooden chopstick through the stem end of each cherry to remove the pits.
Prepare the Crust: Place first 4 ingredients in the large bowl of a food processor; pulse 5 times to combine. Add the butter and shortening. Pulse 12 to 15 times until mixture resembles small peas. Evenly sprinkle 8 tablespoons ice water over the flour and butter mixture. Pulse 5 times to combine. Pinch dough; if it does not stick together, add up to 2 additional tablespoons, 1 tablespoon at a time, and pulse 3 to 5 more times after each addition to combine. Turn dough onto a smooth surface, and knead 4 to 5 times to bring together. Divide in half, and form into 2 rounds, about 1/2 inch thick. Wrap in plastic wrap, and refrigerate 1 to 24 hours.
Preheat oven to 425°F. Remove dough from the refrigerator; let rest 15 minutes. Place 1 round on a lightly floured piece of wax paper; sprinkle dough with flour. Top with another sheet of wax paper. Roll dough to 1/8-inch thickness and 13 inches wide. Place on a baking sheet, and return to refrigerator.
Repeat the rolling process with the second round of dough. Remove and discard top sheet of wax paper. Starting at 1 edge, wrap dough around rolling pin, separating dough from bottom sheet of wax paper as you roll. Discard bottom sheet of wax paper. Unroll dough onto a 9-inch pie pan; gently press dough into pie pan. Chill until ready to use.
Prepare the Filling: Combine the cherries, plums, and vanilla in a large bowl. Mix the sugar, cinnamon, and cornstarch in a small bowl. Combine the dry ingredients with the fruit, and transfer to the prepared pie pan.
Remove the reserved crust from the refrigerator. Cut second dough round into 16 (1/2-inch-wide) strips. Arrange in a lattice design over pie filling. Place pie on a baking sheet. Brush dough with heavy cream, and sprinkle with turbinado sugar. Bake in preheated oven 20 minutes; reduce oven temperature to 350°F, and bake until crust is browned and filling is cooked through, 45 to 50 minutes. Cool to room temperature before serving, about 4 hours.