Preheat oven to 400 degrees Fahrenheit.
In a large skillet melt the butter on medium-high heat. Add in the carrots, celery, and onion, stirring and cooking for 2-3 minutes. Then, mix well and add in the chicken broth. Cook on medium-high heat with lid on for 10 minutes, Stirring occasionally.
While the vegetables cook, roll out one of the pie crust. Place over the top of a pie plate and crimp edges. Poke the bottom of the pie crust with the tines of a fork. Pre-bake crust for 15 minutes until the crust starts to harden and set.
Once the veggies are semi-cooked add the diced mushrooms, green peas, 1 tablespoon poultry seasoning, ½ teaspoon salt and ¼ teaspoon black pepper. Mix well and let simmer for a few minutes.
While vegetables continue to cook, using a small bowl, combine the milk and cornstarch until well blended. Pour the mix into the pot and mix until well incorporated. Cook on medium heat for 5 minutes until the mixture thickens (the sauce will begin to thicken over time). Add the cooked diced/shredded chicken, mix well, and then remove from heat and set to the side.
Remove the semi-cooked pie crust from oven. Carefully, pour the chicken pot pie filling into the pre-baked crust.
Roll out the second pie crust. Place on top of the pot pie. Carefully crimp edges to secure pie crust (remember pie plate is very hot, so be very careful with this step). Cut a few slits in the top of the pie crust. Brush the top of the pie crust with a beaten egg and sprinkle salt and cracked black pepper on top.
Bake for 30 minutes until pie crust is golden brown. Carefully remove from the oven and let sit for 10 minutes. Slice and serve immediately.