Crust
½ cup sour cream, chilled
1 large egg, lightly beaten
2½ cups (12½ ounces) all-purpose flour
1½ teaspoons salt
12 tablespoons unsalted butter, cut into ½-inch pieces and chilled
Pie
4 tablespoons unsalted butter
1 onion, chopped fine
2 carrots, peeled and cut into ¼-inch pieces (2/3 cup)
2 celery ribs, cut into ¼-inch pieces (½ cup)
½ teaspoon salt
½ teaspoon pepper
6 tablespoons all-purpose flour
2¼ cups chicken broth
½ cup half-and-half
1 small russet potato (6 ounces), peeled and cut into ¼-inch pieces (1 cup)
1 teaspoon minced fresh thyme
1 (2½-pound) rotisserie chicken, skin and bones discarded, meat shredded into bite-size pieces (3 cups)
¾ cup frozen peas
1 large egg, lightly beaten
1. 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.
2. 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 disks tightly in plastic wrap and refrigerate for 1 hour. (Wrapped dough can be refrigerated for up to 2 days or frozen for up to 2 months. If frozen, let dough thaw completely on counter before rolling.)
3. Let chilled dough sit on counter to soften slightly before rolling, about 10 minutes. Roll 1 disk of dough into 12-inch circle on lightly floured counter. Loosely roll dough around rolling pin and gently unroll it onto 9-inch pie plate, letting excess dough hang over edge. Ease dough into plate by gently lifting edge of dough with your hand while pressing into plate bottom with your other hand.
4. Roll other disk of dough into 12-inch circle on lightly floured counter, then transfer to parchment paper–lined baking sheet and cover with plastic. Refrigerate both doughs for 30 minutes.
5. For the pie: Meanwhile, adjust oven rack to lowest position and heat oven to 450 degrees. Melt butter in large saucepan over medium heat. Add onion, carrots, celery, salt, and pepper and cook until beginning to soften, about 6 minutes. Add flour and cook, stirring constantly, until golden, 1 to 2 minutes. Slowly stir in broth and half-and-half and bring to boil over medium-high heat.
6. Stir in potato and thyme. Reduce heat to medium and simmer until sauce is thickened and potatoes are tender, about 8 minutes. Off heat, stir in chicken and peas.
7. Transfer chicken mixture to dough-lined pie plate. Loosely roll remaining dough round around rolling pin and gently unroll it onto filling. Trim overhang to ½ inch beyond lip of plate. Pinch edges of top and bottom crusts firmly together. Tuck overhang under itself; folded edge should be flush with edge of plate. Crimp dough evenly around edge of plate using your fingers. Cut four 2-inch slits in top of dough.
8. Brush top of pie with egg. Place pie on rimmed baking sheet. 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, 12 to 15 minutes longer. Let pie cool on wire rack for at least 45 minutes. Serve.
To Make Ahead
Prepare recipe through step 6. Transfer chicken mixture to bowl and refrigerate until fully chilled, about 1½ hours. Continue with step 7, then wrap pie tightly with plastic wrap and then cover with aluminum foil. Freeze for up to 1 month.
Do not thaw before baking. When ready to cook, unwrap pie, cover with foil, and place on rimmed baking sheet. Place on middle rack of cold oven, then set oven to 375 degrees. Bake for 75 minutes. Uncover, brush with egg,and continue to bake until crust is golden brown and filling is beginning to bubble up through slits and registers at least 150 degrees, 55 to 75 minutes. Let cool for 45 minutes before serving.