Preheat the oven to 400°F. Butter the bottom and sides of a 9-inch pie plate. Line the bottom of the pie plate with one of the pie crusts and set aside.

In a large pot (at least 3-quart), add chicken and cover with water by about an inch. Add a pinch of salt and pepper. Bring to a boil, then reduce heat to maintain a gentle simmer and cook for 20 minutes, until the chicken reaches 165°F internally and shreds easily with a fork. Remove chicken from pot and let cool for 5 minutes until safe to handle, then shred into bite-sized pieces.

While the chicken is cooking, melt the butter in a heavy-bottomed skillet over medium heat. Add the diced onion and celery, sautéing for 5 minutes until softened and fragrant. Stir in the garlic and cook for 1 more minute, being careful not to let it brown.

Sprinkle the flour over the sautéed vegetables and stir constantly to create a roux. Cook for 2 to 3 minutes, stirring continuously, until the flour turns light golden brown and smells nutty, similar to toasted bread.

While still stirring, slowly pour in the chicken broth and milk to prevent lumps. Bring the mixture to a simmer and cook for about 4 minutes until it thickens noticeably and coats the back of a spoon.

Stir in the frozen peas and carrots (straight from the freezer), fresh thyme, and shredded chicken. Season with salt and pepper to taste. The filling should be thick and creamy, not runny.

Pour the chicken filling into the prepared pie dish. Lay the second pie crust over the top, trim any excess dough, then press and crimp the top and bottom crusts together with your fingers or a fork to seal completely. Brush the entire top and edges with beaten egg. Use a fork to poke 4 to 5 small holes in the center to release steam.

Bake for 30 to 40 minutes, until the pie crust is deeply golden brown. If the edges brown too quickly, tent them with aluminum foil.

Let the pot pie rest for 10 minutes on a cooling rack before slicing to allow the gravy to set up properly and ensure a crispy bottom crust. Slice and serve warm.