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1
Prepare the Brown Butter Crust: Line a small bowl with aluminum foil. Melt 1 cup plus 2 tablespoons butter in a small heavy saucepan over medium, and cook, stirring constantly, until butter begins to turn golden brown, 6 to 8 minutes. Remove pan from heat, and immediately pour butter into prepared bowl. Freeze until butter is cool and begins to solidify, about 1 hour. Lift foil from bowl; remove butter from the foil, and cut it into small pieces.
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2
Pulse 3 cups flour and 1 1/2 teaspoons salt in a food processor until combined, 3 to 4 times. Add browned butter pieces, and pulse until mixture resembles coarse meal. Stir together 1 egg and 1 tablespoon ice-cold water. With processor running, gradually add egg mixture and up to 5 tablespoons ice-cold water, 1 tablespoon at a time, and process until dough begins to come together. Turn dough out on a lightly floured surface, and knead 3 to 4 times. Divide dough in half. Shape each half into a ball; flatten each ball into a disk, and wrap tightly in plastic wrap. Chill 30 minutes or up to 2 days.
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3
Prepare the Apple Filling: Melt 2 tablespoons of the butter in a large skillet over medium. Add apples, and cook, gently stirring, until apples are just tender, 10 to 15 minutes. Spread apples on a rimmed baking sheet, and cool 30 minutes. Toss together apples, brown sugar, cinnamon, nutmeg, 1 tablespoon flour, and 1/2 teaspoon salt in a large bowl.
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4
Preheat oven to 425u00b0F. Unwrap 1 dough disk, and place on a lightly floured surface. Roll to 1/8-inch thickness. Fit dough into a lightly greased 9-inch deep-dish pie plate or pie pan. Spoon apple mixture into piecrust, spreading and pressing into an even layer. Chop remaining 2 tablespoons butter, and sprinkle over filling.
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5
Repeat rolling procedure with remaining dough disk; cut dough into about 8 (1-inch-wide) strips. Arrange about half of the piecrust strips vertically over the filling. Fold down every other strip, and place 1 strip horizontally across the top of the pie. Unfold the folded strips, and fold up the remaining strips. Lay another strip horizontally across the pie. Continue weaving the remaining strips to form a lattice. Trim excess dough from pie plate, and crimp the edges of the pie. Whisk together cream and 1 egg. Brush entire pie with egg mixture. Sprinkle with granulated sugar. Freeze 30 minutes or until dough is firm. Place pie on an aluminum foil-lined baking sheet.
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6
Bake on lower oven rack of preheated oven 20 minutes. Reduce oven temperature to 375u00b0F, and bake 40 to 45 minutes, shielding pie with foil to prevent over-browning after 20 minutes, if needed, until crust is golden and filling is bubbly. Remove from baking sheet to a wire rack; cool at least 1 hour before serving. Serve with ice cream, if desired.
