Preheat your oven to 425°F (220°C).
In a large saucepan, melt the butter over medium heat.
Add the chopped onion, celery, and carrots; sauté until softened (about 5 minutes).
Stir in the flour and cook for 1-2 minutes.
Gradually whisk in the chicken broth and milk until smooth.
Add the shredded chicken, peas, salt, pepper, and thyme; mix well.
Remove from heat and let the filling cool slightly.
Roll out one pie crust and fit it into a pie dish.
Pour the chicken filling into the crust, spreading it evenly.
Place the second crust over the filling, sealing the edges, and cut slits for steam.
Bake in the preheated oven for 30-35 minutes, or until golden brown.
Let it cool for 10 minutes before serving.