Make the crust: Pulse the flour, granulated sugar and salt in a food processor to combine. Add one-quarter of the butter; process until it disappears into the flour, about 30 seconds. Add the remaining butter and pulse until the mixture looks like cornmeal with bean-size bits of butter. Combine the water and vodka and drizzle over the flour mixture. Pulse until the dough just starts coming together but stop before it gathers into a ball. Divide evenly between 2 pieces of plastic wrap and gently pat each into a disk, using the plastic wrap to help you. (If the dough seems dry with large bits of butter, turn it out onto a work surface and use the heel of your hand to press the dough together.) Wrap tightly and refrigerate until firm, at least 1 hour and up to 24 hours.
Make the filling: Peel and thinly slice the apples; toss in a bowl with the lemon juice. Melt the butter in a large skillet over medium-high heat; add the apples and cook, tossing occasionally, until juicy, about 10 minutes. Remove from the heat; fold in the cherries, flour, extract and allspice. Transfer to a bowl and let cool completely, about 30 minutes.
Roll out 1 disk of dough into a 12-inch round on a lightly floured surface. Ease into a 9-inch pie plate; refrigerate until ready to fill. Place a piece of parchment paper on an inverted baking sheet. Roll out the remaining disk of dough into a 12-inch round, transfer to the parchment and refrigerate until firm, about 30 minutes.
Transfer the dough (on the parchment) to a work surface. Using a fluted pastry wheel or a chef's knife, cut the dough into 14 to 16 half-inch-wide strips. Weave into a lattice on the parchment. Slide back onto the baking sheet and refrigerate until firm, about 30 minutes.
Pour the filling into the crust. Gently slide the lattice off the parchment and onto the fruit, adjusting as needed; trim the excess dough if needed, leaving a 1-inch overhang. Fold the overhanging dough under itself. Brush the lattice with the beaten egg and sprinkle with coarse sugar. Refrigerate the pie 30 minutes.
Position a rack in the lowest position of the oven and preheat to 400 degrees F. Place the pie on a foil-lined baking sheet and bake until the crust is deep golden brown and the filling is bubbling, about 1 1/2 hours. Transfer to a rack and let cool at least 2 hours.
Photograph by Ryan Liebe