Preheat oven to 400°F (200°C).
In a large skillet, melt butter over medium heat.
Add onion, carrots, celery, and peas; sauté until vegetables are tender, about 5 minutes.
Stir in flour and cook for 1-2 minutes to form a roux.
Gradually whisk in chicken broth and milk, stirring constantly until sauce thickens.
Add diced chicken, salt, pepper, and thyme; cook until chicken is heated through and filling is creamy.
Divide the filling among the pie crust dough rounds placed in individual pie tins or ramekins.
Cover each with a second crust if desired, sealing edges and cutting slits for steam.
Bake for 25-30 minutes or until crust is golden brown and filling bubbles.
Let cool slightly before serving on white ceramic plates.