Preheat your oven to 375°F (190°C) for the ideal cooking temperature.
Heat a large skillet over medium heat and melt butter; then add carrots, potatoes, onions, pepper, salt and poultry seasoning. Cook for 8-10 minutes stirring occasionally until vegetables are tender.
Add in the shredded chicken and cook an additional 3-4 minutes until heated through; then remove from heat.
In a separate saucepan over medium heat, whisk together flour and broth to create a smooth mixture then slowly add in milk while continuously whisking until thickened; about 5 minutes. Pour this creamy mixture into the skillet containing chicken and vegetables and stir everything together until it's fully combined; set aside to cool slightly while preparing the crust.
Generously grease a 9-inch pie dish and press one of the pie crusts into it, making sure to fit it nicely against the sides. Fill with chicken and vegetable mixture then scatter frozen peas on top.
Place second crust on top of filling; crimp edges together to form a seal and cut several slits in the center for steam to be released. Brush all over with egg wash then place pie dish into preheated oven and bake for 35 minutes until a golden brown crust has formed; let cool slightly before serving warm.