In a large bowl, whisk together flour, salt, and sugar. Add cold butter cubes and use a pastry cutter or your fingertips to work the butter into the flour until the mixture resembles coarse crumbs with some pea-sized butter pieces remaining.
Sprinkle ice water, one tablespoon at a time, mixing gently with a fork until the dough just comes together. Divide the dough into two equal portions, flatten into discs, wrap in plastic, and refrigerate for at least 1 hour.
In a large mixing bowl, combine the sliced apples and pears. Toss with lemon juice to prevent browning and enhance the natural flavors.
In a separate bowl, mix sugar, cardamom, cinnamon, nutmeg, salt, and cornstarch until well combined. Pour this sugar-spice mixture over the fruit and toss gently until evenly coated. Let the mixture sit for 15-20 minutes to allow the fruit to release some juices.
Preheat your oven to 425°F (220°C) with a baking sheet inside.
Roll out one disc of dough on a lightly floured surface into a 12-inch circle. Transfer to a 9-inch pie dish, gently pressing into the bottom and sides. Trim any excess dough, leaving about a 1-inch overhang.
Pour the fruit filling into the pie crust, mounding slightly in the center. Dot the top with small pieces of butter.
Roll out the second disc of dough into another 12-inch circle. You can either place it whole over the filling or create a lattice top for a more intricate design. If using a whole top crust, be sure to cut several vents to allow steam to escape during baking.
Trim excess dough and crimp the edges to seal. Brush the top with beaten egg and sprinkle with a little sugar for a beautiful golden shine.
Place the pie on the preheated baking sheet and bake at 425°F (220°C) for 20 minutes. Then reduce the temperature to 375°F (190°C) and continue baking for 25-30 minutes more, until the crust is golden brown and the filling is bubbling.
If the edges begin to brown too quickly, cover them with aluminum foil or a pie shield. Allow the pie to cool completely on a wire rack for at least 2 hours before slicing to let the filling set properly.