For the crust: Whisk the flour, sugar and salt together in a medium bowl. Using your fingers, work the butter into the dry ingredients until it resembles coarse cornmeal mixed with pea-sized bits of cold butter. (If the butter gets too soft, refrigerate the mixture for 10 minutes before proceeding.) Use a fork to stir in the egg and water mixture until the dough just comes together. (If the dough is dry, add up to 1 tablespoon more cold water.) Divide the dough into two equal-sized disks, wrap in plastic wrap and refrigerate until thoroughly chilled, at least 1 hour.
For the filling: Peel, halve and core the apples. Cut each half into 4 wedges and place in a bowl with the lemon juice and cranberries, tossing to combine. Add the sugar and toss again to combine evenly.
Melt the butter over medium-high heat in a large skillet. Add the apple mixture and cook, stirring occasionally, until the sugar dissolves and the mixture begins to simmer, about 2 minutes. Cover the pan and reduce the heat to medium-low, cooking until the apples soften and release most of their juices and the cranberries have burst, about 7 minutes.
Strain the apple mixture into a colander set over the bowl, shaking it to collect as much of the juice as possible. Pour the liquid back into the skillet, add the cinnamon and ginger, and simmer over medium heat until thickened and lightly caramelized, about 10 minutes. Return the reduced juice and the apple mixture to the bowl and toss to recombine. Chill the filling until it cools completely, at least an hour and up to 2 days.
To assemble the pie: On a lightly floured surface, roll each disk into an 11- to 12-inch circle. Lay each dough circle between two pieces of parchment or wax paper on a baking sheet and refrigerate for at least 10 minutes.
Place a baking sheet on a rack positioned in the lower third of the oven and preheat to 375 degrees F. Line the bottom of a 9-inch pie dish with one of the dough discs and trim it to leave a 1/2-inch overhang on all sides. Stir the cornstarch into the filling and add it to the pan.
Use a chef's knife, pizza cutter or fluted dough cutter to cut the second round into 1/2-inch thick strips. Lay strips of dough, evenly spaced, across the entire pie. Weave more strips of dough perpendicular through the previous strips to make a lattice or basket weave design across the entire pie. Trim the excess ends from the strips of dough. Pinch the bottom crust edge and lattice edge together, and flute the edge as desired. Brush the surface of the dough with egg and then sprinkle with sugar. Refrigerate for at least 30 minutes.
Bake the pie on the preheated baking sheet until the crust is golden, 50 to 60 minutes. If the edges begin to brown too fast, cover them with strips of aluminum foil. Let the pie cool 1 hour before serving. Once completely cooled, the pie keeps well at room temperature (covered) for 24 hours, or refrigerated for up to 4 days.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.) To make the dough in a food processor: Pulse the flour, sugar and salt in a food processor until combined. Add the butter and pulse until it resembles coarse cornmeal mixed with pea-sized bits of butter, about 10 times. Add the egg and water mixture and pulse 1 to 2 times (do not let the dough form into a ball in the machine). If the dough is very dry, add up to 1 tablespoon more of cold water. Turn the flour mixture out of the machine onto a work surface and bring the dough together by hand.
Copyright 2016 Television Food Network, G.P. All rights reserved.