1/2 cup sour cream, chilled
1 large egg, lightly beaten
2 1/2 cups (12 1/2 ounces) all-purpose flour
1 1/2 teaspoons salt
12 tablespoons unsalted butter, cut into 1/2-inch pieces and chilled
Pie
2 (10- to 12-ounce) bone-in split chicken breasts, trimmed and halved crosswise
3 (5- to 7-ounce) bone-in chicken thighs, trimmed
Salt and pepper
1 tablespoon vegetable oil
3 cups low-sodium chicken broth
1 bay leaf
2 tablespoons unsalted butter
1/4 cup all-purpose flour
1/4 cup half-and-half
1 large egg, lightly beaten
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FOR THE CRUST: Combine sour cream and egg in bowl. Process flour and salt in food processor until combined, about 3 seconds. Add butter and pulse until only pea-size pieces remain, about 10 pulses. Add half of sour cream mixture and pulse until combined, 5 pulses. Add remaining sour cream mixture and pulse until dough begins to form, about 10 pulses.
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Transfer mixture to lightly floured counter and knead briefly until dough comes together. Divide dough in half and form each half into 4-inch disk. Wrap each disk in plastic wrap and refrigerate for at least 1 hour or up to 2 days. (Dough can be wrapped tightly in plastic and foil and frozen for up to 2 months. Thaw completely at room temperature before using.)
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Remove 1 dough disk from refrigerator and let sit for 10 minutes. Working on lightly floured counter, roll into 12-inch round and transfer to 9-inch pie plate, leaving 1/2-inch overhang all around. Repeat with second dough disk and transfer to parchment–lined rimmed baking sheet. Cover both doughs with plastic and refrigerate for 30 minutes.
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FOR THE PIE: Pat chicken dry with paper towels and season with salt and pepper. Heat oil in large Dutch oven over medium-high heat until just smoking. Cook chicken until browned all over, about 10 minutes; transfer to plate. Pour fat (you should have 2 tablespoons) into small bowl; reserve. When chicken is cool enough to handle, remove and discard skin. Add broth, chicken, and bay leaf to now-empty pot and bring to boil. Reduce heat to low and simmer, covered, until breasts register 160 degrees and thighs register 175 degrees, 14 to 18 minutes. Transfer chicken to bowl. When chicken is cool enough to handle, shred into bite-size pieces, discarding bones. Pour broth through fine-mesh strainer into second bowl and reserve (you should have about 2¾ cups); discard bay leaf.
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Adjust oven rack to lowest position and heat oven to 450 degrees. Heat butter and reserved fat in now-empty pot over medium heat until shimmering. Add flour and cook, whisking constantly, until golden, 1 to 2 minutes. Slowly whisk in 2 cups of reserved broth and half-and-half and bring to boil. Reduce heat to medium-low and simmer gravy until thickened and reduced to 1¾ cups, 6 to 8 minutes. Season with salt and pepper to taste. Combine 1 cup gravy with shredded chicken; reserve remaining gravy for serving.
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Transfer chicken mixture to dough-lined pie plate and spread into even layer. Top with second dough round, leaving at least 1/2-inch overhang all around. Fold dough under itself so that edge of fold is flush with outer rim of pie plate. Flute edges using thumb and forefinger or press with tines of fork to seal. Cut four 1-inch slits in top. Brush pie with egg and bake until top is light golden brown, 18 to 20 minutes. Reduce oven temperature to 375 degrees and continue to bake until crust is deep golden brown, 10 to 15 minutes. Let pie cool on wire rack for at least 45 minutes.
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When ready to serve, bring remaining ¾ cup reserved gravy and remaining ¾ cup reserved broth to boil in medium saucepan. Simmer over medium-low heat until slightly thickened, 5 to 7 minutes. Season with salt and pepper to taste. Serve pie with gravy.
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