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Position two oven racks in the lower third of the oven and heat the oven to 400°F.
Peel the apples, cut each in half from top to bottom, remove the cores with a melon baller, and trim the ends with a paring knife. Lay the apples, cut side down, on a cutting board. Cut the Cortland apples (below left) crosswise into 3/4-inch pieces, and then halve each piece diagonally. Cut the Granny Smith apples (below right) crosswise into 1/4-inch slices, leaving them whole. Put the apples in a large bowl and toss with the lemon juice.
Combine the brown sugar, 1/4 cup of the granulated sugar, cornstarch, cinnamon, kosher salt, and nutmeg in a small bowl. (Don’t add this to the fruit yet.)
In a small dish, lightly beat the egg white with 1 teaspoon water. Set aside.
Butter a 9-inch ovenproof glass (Pyrex) pie plate, including the rim, with the 2 tsp. of softened butter.
Rub 2 to 3 Tbs. of flour into the surface of a pastry cloth, forming a circle about 15 inches across, and also into a rolling pin stocking. If you don’t have a pastry cloth, rub the flour into a large, smooth-weave, cotton kitchen towel and use a floured rolling pin. Roll one of the disks of dough into a circle that’s 1/8 inch thick and about 15 inches across.
Lay the rolling pin across the upper third of the dough circle; lift the pastry cloth to gently drape the dough over the pin and then roll the pin toward you, wrapping the remaining dough loosely around it. Hold the rolling pin over the near edge of the pie plate. Allowing for about a 1-inch overhang, unroll the dough away from you, easing it into the contours of the pan. If the dough isn’t centered in the pan, gently adjust it and then lightly press it into the pan. Take care not to stretch the dough. If it tears, simply press it back together—the dough is quite forgiving.
Brush the bottom and sides of the dough with a light coating of the egg-white wash (you won’t need all of it). Leaving a 1/4-inch overhang, cut around the edge of the dough with kitchen shears.
Combine the sugar mixture with the apples and toss to coat well. Mound the apples in the pie plate, rearranging the fruit as needed to make the pile compact. Dot the apples with the 1 Tbs. cold butter cubes.
Rub another 2 to 3 Tbs. flour into the surface of the pastry cloth and stocking. Roll the remaining dough into a circle that’s 1/8 inch thick and about 15 inches across. Use the rolling pin to move the dough. As you unroll the dough, center it on top of the apples. Place your hands on either side of the top crust of the pie and ease the dough toward the center, giving the dough plenty of slack. Leaving a 3/4-inch overhang, trim the top layer of dough around the rim of the pie plate. Fold the top layer of dough under the bottom layer, tucking the two layers of dough together. Press a lightly floured fork around the edge of the dough to seal it, or flute the edge of the dough with lightly floured fingers.
Lightly brush the top with cold water and sprinkle the surface with the remaining 1 Tbs. sugar. Make steam vents in the dough by poking the tip of a paring knife through it in a few places; it’s important to vent well so that the steam from the cooking apples won’t build up and crack the top of the crust.
Cover the rim of the pie with aluminum foil bands. This will prevent the edge of the crust from overbrowning.
Place a rimmed baking sheet or an aluminum foil drip pan on the oven rack below the pie to catch any juices that overflow during baking. Set the pie on the rack above.
Bake until the top and bottom crusts are golden brown and the juices are bubbling, 60 to 75 minutes; to thicken, the juices must boil, so look for the bubbles through the steam vents or through cracks near the edges of the pie and listen for the sound of bubbling juices. During the last 5 minutes of baking, remove the foil bands from the edges of the pie. Cool the pie at least 3 hours and up to overnight before serving.