Preheat the Oven: Set your oven to 425°F (220°C).
Cook the Chicken and Vegetables: In a medium saucepan, combine the cubed chicken, carrots, celery, and peas. Add enough water to cover, bring to a boil, and cook for 10 minutes. Drain and set aside.
Prepare the Sauce: In a large pan, melt the butter over medium heat. Add the onion and garlic, cooking until softened (about 5-7 minutes). Stir in the flour, salt, pepper, and thyme. Slowly whisk in the chicken broth and half-and-half. Reduce the heat and simmer for 5-10 minutes, stirring until the sauce thickens. Adjust seasoning as needed.
Assemble the Pie: Place one pie crust into your deep-dish pie plate. Spread the cooked chicken and veggies evenly over the bottom. Pour the sauce over the top.
Seal and Eggwash: Cover with the second pie crust, seal the edges, and cut a few slits in the top for steam to escape. Brush with the egg wash.
Bake: Bake for 30–35 minutes, or until the crust is golden brown and the filling is bubbling. If the edges brown too quickly, cover them with foil after 20 minutes. Let the pie rest for 10 minutes before serving.