Step 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.

Step 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.)

Step 3

Let chilled dough sit on counter to soften slightly, about 10 minutes, before rolling. 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.

Step 4

Roll other disk of dough into 12-inch circle on lightly floured counter, then transfer to parchment paper–lined baking sheet; cover with plastic. Refrigerate both doughs for 30 minutes.

Step 5

FOR THE FILLING: 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 vegetables begin 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.

Step 6

Stir in potato and thyme. Reduce heat to medium and simmer until sauce is thickened and potato is tender, about 8 minutes. Off heat, stir in chicken and peas.

Step 7

Transfer filling to dough-lined pie plate. Loosely roll remaining dough round around rolling pin and gently unroll it onto filling. Trim overhang to 1/2 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.

Step 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, rotate sheet, 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.

Step 9

TO MAKE AHEAD: At end of step 6, transfer filling to bowland refrigerate until fully chilled, about1 1/2 hours. Continue with step 7, thenwrap pie tightly in plastic wrap andthen aluminum foil. Freeze for up to 1month. When ready to bake, unwrapfrozen pie, cover with foil, and place onrimmed baking sheet (do not thaw).Place sheet on middle rack of cold ovenand set oven to 375 degrees. Bake for1 1/4 hours. Uncover pie and brush withegg. Rotate sheet and continue to bakeuntil crust is golden brown and filling isbeginning to bubble up through slits andregisters at least 150 degrees, 55 minutesto 1 1/4 hours longer. Let cool for 45minutes before serving.