Heat the oven. Set a rack in the lower third, place a baking sheet on it, and heat to 425 degrees F while you fit one pie crust into a 9 inch deep dish plate.
Soften the onion. Melt 1/3 cup butter over medium heat, stir in the onion, and cook about 3 minutes until translucent and tender.
Make the roux. Sprinkle over the flour, salt, pepper, and poultry seasoning, and whisk for 1 minute until a thick paste forms.
Thicken the sauce. Slowly whisk in the broth and milk, then simmer 4 to 5 minutes until glossy and thick enough to coat a spoon.
Combine and fill. Fold in the chicken and vegetables, cool 5 minutes, and scoop the filling into the lined plate.
Cover and bake. Lay the second crust on top, crimp, cut 4 vents, and bake 35 to 40 minutes, shielding the rim with foil for the last 15 minutes, until golden and 165 degrees F in the center. Rest 5 minutes before cutting.