For the pear and fig filling: Preheat the oven to 400 degrees F. Line 2 baking sheets with parchment.
Combine the pears, fig preserves, cornstarch, cinnamon and a pinch of salt in a large bowl. Set aside.
For the apple cranberry filling: Place a large nonstick pot over medium heat. Add the apples, frozen cranberries, sugar and a pinch of salt. Cook, stirring aggressively to burst the cranberries as they soften, about 5 minutes. Meanwhile, whisk the cranberry juice and cornstarch together in a small bowl to form a slurry. Pour the slurry into the fruit and stir to combine, then cook until a thick sauce has formed, 1 minute more. Turn off the heat and set aside.
For the blueberry filling: Combine the blueberries, honey, cornstarch and a pinch of salt in a mixing bowl and stir to combine. Set aside.
For the pie: On a lightly floured surface, stack 3 rounds of pie dough, then roll out to a 20-by-15-inch rectangle. Place the rectangle on one of the parchment lined baking sheets. Use scissors to trim to a 1-inch overhang on all 4 sides and save any excess dough for later use. Place in the fridge to firm up for 20 minutes.
Meanwhile, stack the remaining 3 rounds of dough, then roll out to an 18-by-13-inch rectangle. Use a pizza cutter or knife to slice the rectangle lengthwise into twelve 1-inch-wide strips. Place the strips on the remaining lined baking sheet and chill in the refrigerator for 10 minutes, or until it is time to create the lattice.
Remove the bottom crust from the refrigerator and prick in several places with the tines of a fork. Pour the blueberry filling in one corner and spread diagonally to fill a third of the pan (the fruit layers should not have a perfect edge or stopping point). Next, pour the apple cranberry filling in the middle and spread diagonally to fill another third of the pan. Finally, pour the pear and fig filling in the remaining third of the pan.
Using a basket weave method, create a diagonal lattice on top of the pie with the remaining dough strips: Arrange 6 strips down diagonally across the pie, leaving about 2 inches between each dough strip. Fold back every other strip and place a strip down perpendicularly. Unfold the strips over them. Repeat this process with the remaining 5 strips, alternating the strips that are lifted to create a woven pattern. Press the end of each dough strip along the rim of the baking sheet and trim off any excess. Use your fingertips or the tines of a fork to crimp the crust of the pie along the rim of the pan. Brush the lattice (but not the edges) with the beaten egg.
Bake until the filling is bubbling and the crust is golden brown, 40 to 45 minutes. Let cool completely before slicing.