Preheat your oven to 400°F (200°C).
In a large skillet, melt the butter over medium heat. Add the onion, carrots, and celery. Sauté for about 5–7 minutes until softened. Stir in the garlic and cook for another 30 seconds.
Sprinkle the flour over the vegetables and stir well. Cook for 1 minute to remove the raw flour taste.
Slowly whisk in the chicken broth, followed by the milk. Continue stirring until the sauce thickens, about 3–5 minutes. It should coat the back of a spoon.
Stir in the chicken, peas, thyme, salt, and pepper. Remove from heat and let the filling cool slightly.
Roll out one pie crust and fit it into a 9-inch pie dish. Pour in the chicken filling, spreading evenly.
Place the second pie crust on top, sealing the edges by pinching or crimping. Cut a few small slits in the center to let steam escape.
Brush the top crust with beaten egg for a shiny golden finish.
Bake for 35–40 minutes until the crust is golden brown and the filling is bubbling.
Let rest for 10 minutes before slicing and serving. This helps the filling set.