For the pie crust: Place the flour, granulated sugar and salt in a food processor. Pulse once or twice to evenly incorporate. Add the butter and lard and pulse until the fat is broken down into pea-size pieces, 5 to 6 pulses. Add the cider vinegar and 1 tablespoon of the cold water and pulse to combine. Continue adding a tablespoon of water at a time and pulsing after each addition until the dough just comes together (you may not need the full 1/3 cup water). Remove the dough from the food processor and form a third of the dough (about 9 ounces) into a disk. Form the remaining two-thirds of the dough (about 1 pound) into another disk. Wrap the disks in plastic wrap and refrigerate at least 2 hours or up to overnight.
Position an oven rack in the lower third of the oven and preheat to 425 degrees F.
Remove the dough disks from the refrigerator and let sit at room temperature for 10 to 20 minutes to take the chill off.
For the filling: Combine the brown sugar, granulated sugar, cornstarch, cinnamon, salt, ginger and nutmeg in a small bowl. Set aside.
Put the apple slices in a large bowl and stir in the sugar mixture and orange juice. Mix to combine.
Dust a clean work surface with flour and roll out the larger disk of dough into a 13-inch round. Transfer to a 10-inch pie dish. Pour the apple filling into the shell and then evenly distribute the butter pieces over the apples. Roll out the second disk of dough into an approximately 11-by-10-inch rectangle. Cut the dough into six strips, about 11 inches long by 1 1/4 inches thick.
Form a lattice top by laying 3 parallel dough strips vertically across the pie. Fold the middle strip of dough in half back over itself and lay one piece of dough horizontally across the middle of the pie, perpendicular to the other dough strips. Fold the original strip of dough back down. Next, lift the other two strips of dough and fold them back over themselves. Lay another strip of dough perpendicular to the folded-over dough pieces. Fold those pieces back down and repeat on the other side with the remaining piece of dough. Trim the dough around the edges of the pie dish so there is no more than a 1/2-inch overhang.
Fold the bottom crust up and over the edges of the lattice top and crimp the edges together between your forefingers and thumbs. Place the pie on a baking sheet and transfer to the oven. Bake for 20 minutes. Remove the pie from the oven and cover the edges with foil or a pie crust shield. Return to the oven and continue to bake until the filling begins to bubble, another 35 to 40 minutes. Let cool at least 1 hour.
For the mascarpone whipped cream: Combine the mascarpone, cream and confectioners' sugar in the bowl of a stand mixer fitted with a whisk attachment. Beat on high speed until the mixture forms soft peaks and is thick. Serve immediately or store in the refrigerator until ready to serve. (This can be made the day before.)
Cut the pie into slices and top with the mascarpone whipped cream.