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Remove one of the chilled disks of dough from the refrigerator 10 minutes before rolling it out to a 12"-diameter disk. Line a 9" pie pan with the dough, and place it in the refrigerator while you prepare the filling.
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To make the filling: Peel, core, and slice the pears 1/8" thick into a large mixing bowl. Pit the cherries, halve them, add to the pears, and set aside.
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In a small bowl, whisk together the sugars, flour, cardamom, cinnamon, and salt until well combined. Add the flour-sugar mixture to the pears and cherries, and mix until just combined. Add the lemon juice and vanilla. Mix until the fruit is uniformly coated.
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Spoon the filling into the lined pie pan, mounding it in the center. Dot the top with the tablespoon of butter, and return the pie to the refrigerator while you roll out the top crust with the second disk of dough. You can make a lattice top, or simply place a 12" circle of dough on top of the filling. Tuck the edges under the bottom crust, and pinch the edges of the bottom and top crust together. Cut 3 or 4 steam vents in the center of the top crust. Refrigerate the pie for 1 hour (or freeze for 30 minutes) before baking.
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To bake: Place a rack in the bottom third of the oven and preheat it to 425°F.
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Beat together the egg yolk and heavy cream. Remove the pie from its chilling place and brush the top evenly with the egg wash. Sprinkle with sugar.
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Bake the pie on a baking sheet for 15 minutes, or until the crust begins to turn golden. Reduce the heat to 350°F and bake for 40 to 45 minutes more, until the juices are bubbling through the vents. Cover the edges with a pie shield or foil if they brown too quickly.
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Remove the pie from the oven and cool it on a rack for at least 1 hour before serving.