Step 1

Mix flour, salt, and baking powder in a food processor. Add cubed butter and shortening and process until mixture resembles coarse meal with some pea-size lumps. With machine running, add 1/2 cup ice water and process, adding more water by teaspoonfuls if dry, until dough forms. Form dough into a ball; flatten into a disk. Wrap in plastic and refrigerate for 30 minutes.

Step 2

Line a baking sheet with parchment paper. Place chilled dough on a lightly floured surface and roll out to a 12x9 1/2" rectangle, about 1/4" thick. Place on prepared sheet and cover tightly with plastic wrap. Chill. DO AHEAD: Can be made 6 hours ahead. Keep chilled.

Step 3

Preheat oven to 400°. Place carrots, onion, and leek on a rimmed baking sheet and roast until golden brown, about 30 minutes. Transfer to a large pot fitted with a lid. Add chicken, breast side down, celery, thyme sprigs, wine, salt, and peppercorns to pot. Add 8 cups water to cover and bring to a boil. Reduce heat to low and simmer gently for 10 minutes. Remove pot from heat, cover, and poach chicken until cooked through, about 35 minutes.

Step 4

Remove chicken from pot and set aside until cool. Reserve broth. Shred meat; discard skin and bones. Set a large strainer over another pot. Strain broth into clean pot. Stir in tomato paste. Return to medium heat and simmer, uncovered, until reduced to 5 cups, about 40 minutes. DO AHEAD: Can be made 1 day ahead. Chill uncovered until cool. Cover chicken and broth separately; keep chilled. Rewarm broth before continuing.

Step 5

Place dried chanterelles in a small bowl and cover with 1 cup hot water. Let steep for 10 minutes; set aside. Reserve soaking liquid. Cook carrots in a large saucepan of boiling salted water until crisp-tender, about 2 minutes. Using a slotted spoon, transfer to a medium bowl. Add peas to saucepan; cook for 1 minute; transfer to bowl with carrots. Add onions to saucepan and cook until crisp-tender, about 2 minutes. Remove from pot and set aside; let cool. Peel onions.

Step 6

In a large heavy saucepan, melt butter over medium heat. Whisk in flour. Cook over medium-low heat, whisking constantly, until golden brown, about 10 minutes. Whisk in herbs and warm broth from chicken. Using a slotted spoon, transfer chanterelles to saucepan. Pour in soaking liquid, leaving sediment behind. Simmer for 20 minutes.

Step 7

Set a strainer over another large pot. Strain broth into pot; discard solids. Add chicken, carrots, peas, onions, mushrooms, and potatoes. Bring to a simmer. DO AHEAD: Can be made 1 day ahead. Chill uncovered until cool; cover and keep chilled. Rewarm before continuing.

Step 8

Preheat oven to 400°. Spoon filling into an 11x8x2" or 2 1/2-qt. round baking dish. (Add pie bird, if using.) Top with pastry, pinching edges to seal. If not using pie bird, cut a 1" slit in center of crust for steam to vent. Brush pastry with beaten egg, and sprinkle with sea salt and pepper, if desired.

Step 9

Place pot pie on a rimmed baking sheet. Bake until crust is golden brown and filling is bubbly, about 45 minutes.