Prepare the Pie Crust:

In a large mixing bowl, whisk together the flour and salt. Add the cold butter cubes and use a pastry cutter (or food processor) to cut the butter into the flour until the mixture resembles coarse crumbs. Gradually add the ice water, 1 tablespoon at a time, mixing until the dough comes together. You may need more water. Split the dough into two balls, wrap each in plastic wrap, and refrigerate for at least 30 minutes.

Make the Filling:

In a medium saucepan, melt the butter over medium heat. Add the diced onion and cook for 3–4 minutes until softened. Add the garlic and cook for another minute until fragrant. Stir in the flour and cook for 1–2 minutes to make a roux.

Add the Liquids:

Gradually whisk in the chicken broth and milk, ensuring there are no lumps. Bring to a simmer and cook for 5–6 minutes, stirring frequently, until the sauce thickens.

Combine with Chicken and Vegetables:

Add the cooked chicken, peas, carrots, thyme, salt, and pepper to the sauce. Stir until well combined. Remove from heat and set aside.

Assemble the Pie:

Preheat your oven to 375°F (190°C). Roll out one ball of pie dough on a floured surface into a 12-inch circle. Place it into a 9-inch pie dish, pressing it into the bottom and sides. Pour the chicken mixture into the crust.

Top the Pie:

Roll out the second ball of dough into a 12-inch circle and place it over the filling. Trim any excess dough and crimp the edges to seal the crust. Cut a few slits in the top crust to allow steam to escape.

Apply the Egg Wash:

Beat the egg and brush it over the top crust to create a golden, glossy finish.

Bake the Pie:

Bake in the preheated oven for 45–50 minutes, or until the crust is golden brown and the filling is bubbling. If the edges of the crust begin to brown too quickly, cover them with aluminum foil.

Serve:

Let the pie rest for 10 minutes before slicing and serving. Enjoy!