Step 1: In a large skillet, melt butter over medium heat. Add diced onion and cook until softened, about 3-4 minutes. Sprinkle flour over the onions and stir to form a paste. Cook for 1 minute.
Step 2: Gradually whisk in chicken broth and milk, stirring constantly until the mixture thickens into a creamy sauce. Season with salt, pepper, and thyme. Stir in cooked chicken, peas and carrots, and potatoes. Remove from heat and let the filling cool slightly.
Step 3: Preheat your oven to 400°F (200°C). Place one pie crust into a 9-inch pie dish, pressing it gently to fit. Pour the cooled filling into the crust, spreading it evenly. Cover with the second pie crust, sealing the edges by pinching or crimping. Cut small slits in the top crust to allow steam to escape.
Step 4: Brush the top crust with the beaten egg for a golden finish. Bake in the preheated oven for 30-35 minutes, or until the crust is golden brown and the filling is bubbling. Let the pie cool for 10-15 minutes before serving.