Place the whole chicken in a large pot. Add 1 teaspoon salt, 1/2 teaspoon pepper, quartered onion, chopped carrot, and celery. Cover with water.
Bring to a boil, then reduce to a gentle simmer. Cook 60–75 minutes, or until fully cooked.
Remove chicken and let cool. Strain broth and reserve. Shred/chop chicken and measure 3 cups for the recipe. Reserve 2 cups broth.
In a large skillet, melt 4 tablespoons butter over medium heat. Add diced onion and cook 3–4 minutes until softened.
Sprinkle in 1/3 cup flour and stir constantly for 1 minute.
Slowly whisk in 2 cups broth, then whisk in 1 cup milk. Stir 3–5 minutes until thick and creamy.
Stir in 1 teaspoon salt, 1/2 teaspoon pepper, 1/2 teaspoon thyme, and 1/4 teaspoon poultry seasoning.
Add 3 cups cooked chicken and 2 cups mixed vegetables. Stir until combined. Remove from heat and let cool 5–10 minutes.
Preheat oven to 375°F. Lightly grease a 9 x 11-inch baking dish.
Roll out one crust and press it into the dish. Trim overhang as needed.
Spoon filling into crust and spread evenly.
Roll out second crust and place on top. Trim excess, seal edges, and cut 4–6 small slits to vent.
Optional: brush with egg wash.
Bake on a baking sheet for 45–50 minutes, or until crust is golden and filling is bubbling.
Rest 10–15 minutes before slicing or scooping.