Prepare the filling: Peel, halve and core the apples. Cut each half into 6 wedges and toss with the sugar and lemon juice in a bowl. Melt 4 tablespoons butter in a large skillet over medium-high heat. Add the apple mixture and cook, stirring occasionally, until the firmer apples begin to soften but still hold their shape, 8 to 9 minutes. Sprinkle with the flour, cinnamon, cloves and nutmeg and toss until the juices begin to thicken, about 1 minute. Remove from the heat and cool completely. (The filling can be made up to 2 days ahead; cover and refrigerate.)
Make the dough: Pulse the flour, sugar and salt in a food processor until combined. Add the shortening and pulse until it disappears, about 30 seconds. Add the butter and pulse until the mixture looks like coarse meal with pea-size bits of butter. Beat 1 egg with 1/4 cup ice water in a bowl, then add to the processor and pulse once or twice. (Stop before the dough gathers into a ball.) Lay out 2 sheets of plastic wrap; divide the dough between the sheets and pat each into a disk. Wrap the disks tightly and refrigerate until firm, at least 1 hour.
Roll 1 disk of dough into a 12-inch round on a floured surface (or between 2 pieces of floured parchment paper). Ease into a 9 1/2-inch deep-dish glass pie plate. Add the filling, mounding it slightly, and dot with the remaining 2 tablespoons butter. Chill the pie while you roll the remaining dough into a 12-inch round.
Lay the dough over the filling and press the 2 crusts together around the edges. Fold the overhanging dough under itself and crimp with your fingers. Beat the remaining egg in a bowl and brush over the top crust; sprinkle with sugar. Pierce the top with a knife a few times to let steam escape. Chill 30 minutes.
Preheat the oven to 375 degrees F. Place the pie on a baking sheet; bake until the crust is golden and the filling is bubbly, 55 to 60 minutes. (Cover the edges with foil if they brown too quickly.) Transfer to a rack and cool until set, 2 hours.