Preheat the oven to 350 degrees F. Roll out 1 piece of dough into a 12-inch round on a lightly floured surface. Ease into a 9-inch pie plate; chill until ready to use.
Make the filling: Peel and chop the apples. Pulse the apples, raisins, and dried plums, figs and cherries in a food processor until chunky. Transfer to a large bowl and toss with the sugar, tapioca, orange liqueur, lemon zest, cinnamon, salt and allspice. Spoon into the crust and dot with the butter.
Roll out the second piece of dough into a 12-inch round on a lightly floured surface. Cut into 1/2-inch-wide strips with a knife. Arrange half of the strips in parallel rows over the filling, about 1/2 inch apart. Working with 1 strip at a time, arrange the remaining strips perpendicular to the first ones, weaving the strips over and under, to form a lattice. Fold the overhanging dough under itself and crimp the edge with your fingers. Brush the lattice crust with the beaten egg.
Put the pie on a rimmed baking sheet and transfer to the oven; bake until the crust is golden and the filling is bubbly, about 1 hour. (Cover the edge with foil if it is browning too quickly.) Transfer to a rack and let cool until set, about 3 hours.