Preheat oven to 375 °F.
Place chicken, whole onion half and bay leaves in a large pot and cover with cold water. Place pot over medium-high heat and bring up to a simmer. Reduce heat to medium and simmer until chicken has cooked through, about 20 minutes. Remove chicken from pot and let cool. Reserve about 1 cup (a mugful) of the cooking liquid and then discard the remaining.
While chicken cools, place a large Dutch oven over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Add chopped onion, mushrooms, carrots, celery and parsnips to the pan, season with salt and pepper, and cover. Cook, stirring occasionally, until softened, 7-8 minutes. While the veggies are cooking, remove the chicken breast meat from the bones and shred.
When the veggies are tender, scoot them over to the side of the pan and add the butter to the center of the pot. Once butter has melted, add the flour and cook for about a minute. Whisk in white wine, milk, chicken stock and reserved poaching liquid. Bring up to a simmer then add the tarragon, Dijon, peas and shredded chicken. Bring back up to a simmer, season with salt and pepper, and cook until the sauce has thickened, 2-3 minutes.
Transfer to a casserole dish and cover with the pastry dough, trimming it as needed to cover the entire surface. Make a few slits in the dough to allow steam to escape, then brush the top with the beaten egg.
Bake the pot pie until the filling is bubbling and the crust is golden brown, about 30 minutes. Serve.
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