Instructions

to make the dough

Process flour and salt together in food processor until combined, about 3 seconds. Add butter and pulse until size of large peas, about six to eight 1-second pulses.
Mix sour cream and 4 tablespoons ice water in small bowl until combined. Add half of sour cream mixture to flour mixture; pulse for three 1-second pulses. Repeat with remaining sour cream mixture. Pinch dough with fingers; if dough is floury, dry, and does not hold together, add 1 to 2 additional tablespoons water and process until dough forms large clumps and no dry flour remains, three to five 1-second pulses.
Turn dough onto work surface. Shape into ball and flatten to 5-inch disk; wrap in plastic and refrigerate until firm but not hard, 1 to 2 hours. (Dough can be refrigerated overnight.)

to roast chicken and vegetables

Adjust oven racks to lower-middle and upper-middle positions; heat oven to 450 degrees. Heat oil in heavy-bottomed 12-inch ovenproof skillet over medium-high heat until oil just begins to smoke; swirl skillet to coat evenly with oil. Brown chicken pieces skin side down until deep golden, 3 to 4 minutes; turn chicken pieces and brown until golden on second side, 3 to 4 minutes longer. Transfer chicken to large plate; drain all but 1 tablespoon fat.
Set skillet over medium heat. Add pearl onions and cook, stirring occasionally, until browned, about 3 minutes. Increase heat to medium-high and add onion, carrots, celery, and salt and pepper to taste; cook, stirring occasionally, until vegetables begin to brown, 2 to 3 minutes. Return chicken pieces to skillet skin side up and place in oven on lower-middle rack. Roast until thickest part of breast registers about 160 degrees on instant-read thermometer and thickest part of thigh and drumstick registers about 175 degrees, 18 to 25 minutes. Using potholder or oven mitt, remove skillet from oven. Transfer chicken to platter and let rest 1 hour. Set skillet with vegetables aside.

to roll out the dough

(If dough has been chilled longer than 2 hours, let stand at room temperature for 15 minutes before rolling.) Dust 16 by 12-inch sheet of parchment paper liberally with flour; roll, flute, and cut vent holes according to illustrations above. Transfer parchment and dough to baking sheet; chill in freezer until ready to use, at least 30 minutes (or refrigerate for 1 hour).

to make sauce and assemble pie

While chicken rests, return skillet with vegetables to medium-high heat. Add chicken broth and any juices from chicken platter and bring to simmer, scraping sides and bottom of skillet with wooden spoon to loosen any browned bits. Pour contents of skillet through medium-mesh strainer set over large bowl; transfer vegetables from strainer to separate bowl and refrigerate. After fat rises to surface of broth, 15 to 20 minutes, skim with ladle or large spoon and discard. (You should have about 3 1/2 cups broth.)
When chicken has rested, remove and discard skin. Using fingers or fork, pull chicken off bones into 2-inch shreds and 1-inch chunks; refrigerate until ready to use (you should have about 4 1/2 cups chicken). Wipe skillet clean; melt butter over medium-high heat until foaming. Stir in flour and cook, stirring constantly, until mixture darkens slightly and becomes fragrant, about 1 minute. Slowly whisk in broth and cream and bring to boil; reduce heat to medium and simmer, stirring constantly, until sauce is thickened and coats back of spoon, 8 to 10 minutes. Off heat, add chicken, reserved vegetables, peas, lemon juice, parsley, thyme, and salt and pepper to taste; gently stir to combine. Remove dough from freezer, lift off parchment paper, and place dough on top of filling. Bake on upper-middle rack at 450 degrees until crust begins to brown around edges, about 15 minutes; reduce heat to 375 degrees and continue to bake until crust is golden brown and filling bubbles, 15 to 20 minutes more. Let cool 15 minutes before serving.