Make the crust: Whisk the flour, granulated sugar and salt in a large bowl. Toss the butter cubes in the flour mixture, then press the cubes between your fingertips until you have large, thin flour-coated flakes. Mix the ice water and vodka together, then drizzle into the flour mixture; stir with a fork until evenly moistened and clumps start forming. Gather the dough together into a ball, then press and fold a few times until fully combined. Divide the dough between 2 pieces of plastic wrap; flatten into disks and wrap tightly. Refrigerate until firm, at least 1 hour or overnight.
Meanwhile, make the filling: Toss the apples, granulated sugar, lemon juice and salt in a large bowl. Melt the butter in a large skillet over medium-high heat. Stir in the apple mixture and bring to a simmer. Cover and cook, stirring occasionally, until the apples are tender, release their juices and begin to shrink, about 12 minutes. Stir together the cherry juice and cornstarch in a small bowl, then stir into the skillet along with the cherries and return to a simmer. Cook, uncovered, until thick and glossy, 1 to 2 minutes. Transfer the filling to a bowl and refrigerate until completely cool, at least 1 hour or up to 1 day.
Roll out 1 disk of dough on a lightly floured surface into a 14-inch round. Ease into a 9-inch pie plate. Trim the edges, leaving a 1-inch overhang. Brush off any excess flour and refrigerate. Roll out the other disk of dough into a 13-inch round. Transfer the dough to a parchment-lined baking sheet and refrigerate along with the crust until very firm, at least 1 hour.
Scrape the filling evenly into the chilled pie crust. Cut 10 strips of varying widths from the round of dough (on the parchment) with a fluted pastry cutter. Lay 5 strips of dough across the pie, leaving space between each. Lay 5 more strips across the pie in the opposite direction, weaving them through to make a lattice. Trim the strips, leaving a 1-inch overhang, then fold the overhanging dough (strips and bottom crust) under itself and crimp with your fingers. Refrigerate the pie to firm up the pastry again, 1 hour.
Place a foil-lined baking sheet on the middle oven rack; preheat to 375˚. Brush the pie crust (lattice and edge) with the beaten egg and sprinkle with coarse sugar. Bake the pie on the hot baking sheet until the crust is golden and the filling is bubbling, about 1 1/2 hours. Transfer the pie to a rack to cool completely.