Place whole chicken in a large pot, covering with 2 quarts of water. Add onion, celery, bay leaf, and 1 tsp of salt. Bring to a boil, then simmer for 1 hour until chicken is tender. Remove chicken and let cool, reserving the broth.

Skim fat from reserved broth and return it to the pot. Bring to a simmer and add mixed vegetables and cubed potatoes; cook for 8 minutes until tender. Remove from heat and set aside to cool slightly.

Melt ½ cup of butter in the pot over medium heat. Sprinkle in flour, stirring to create a roux. Gradually whisk in reserved broth and 1 cup of milk, stirring until thickened (about 5-7 minutes). Season with 1½ tsp salt, 1¼ tsp black pepper, and ¼ tsp thyme.

Stir shredded chicken, cooked vegetables, and sliced eggs into the sauce until well combined. Pour into a greased 9-inch baking dish and distribute evenly.

Unroll pie crust over filling, pinch and seal edges, and cut slits in top for steam. Bake in preheated oven at 400°F for 20 minutes until crust is golden brown.