Preheat your oven to 375ºF.
In a large pot, melt the butter over medium-high heat.
Add the onion, carrot, and celery.
Cook, stirring occasionally, until the onion is translucent, about 3 minutes.
Stir in the shredded chicken.
Sprinkle the flour over the mixture and stir until it’s all combined with the meat and vegetables.
Gradually add the chicken broth, stirring constantly.
Continue to simmer until the mixture thickens, about 5-7 minutes.
Add the peas, salt, and black pepper.
Transfer the chicken mixture to a pie dish.
Cover with the pie crust, trimming any overhanging dough.
Brush the crust with the beaten egg and use a knife to make a few small steam vents.
Bake in the preheated oven until the crust is golden brown, about 24-30 minutes.