Preheat Oven:
Preheat to 400°F (200°C).
Cook the Vegetables:
In a large skillet, melt butter over medium heat.
Add onion, garlic, carrots, and celery. Cook for 5–6 minutes until slightly softened.
Make the Sauce:
Sprinkle in flour and stir constantly for 1–2 minutes to form a roux.
Gradually whisk in chicken broth and milk until smooth and thickened (about 3–5 minutes).
Add the Chicken and Peas:
Stir in the chicken, peas, salt, pepper, thyme, and optional seasonings.
Remove from heat and let cool slightly while you prep the crust.
Assemble the Pie:
Place one crust into the bottom of a 9-inch pie dish.
Pour the filling evenly into the crust.
Cover with the second crust. Trim excess dough and crimp edges to seal.
Cut 3–4 small slits in the top to vent. Brush with beaten egg for a golden finish.
Bake:
Bake for 35–40 minutes, or until crust is golden brown and filling is bubbling.
If the edges brown too quickly, cover with foil strips during the last 10–15 minutes.
Cool & Serve:
Let rest for 10–15 minutes before slicing and serving.