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. Can be made 1 day ahead. Chill uncovered until cool. Cover chicken and broth separately; keep chilled. Rewarm broth before continuing.
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.
Place dried mushrooms in a small bowl and cover with 1 cup of 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.
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 mushrooms to saucepan. Pour in soaking liquid, leaving sediment behind. Simmer for 20 minutes.
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. Can be made one day ahead. chill uncovered until cool; cover and keep chilled. Rewarm before continuing.
Preheat oven to 400 degrees. Spoon filling into an 11x8x2" or a 2 1/2 quart round baking dish. Top with pastry, pinching edges to seal. Cut a 1" slit in center of crust for steam to vent. Brush pastry with beaten egg, and sprinkle with sea salt and pepper.
Place pot pie on a rimmed baking sheet. Bake until crust is golden brown and filling is bubbly, about 45 minutes.