Preheat your oven to 375°F (190°C).

In a large skillet over medium heat, sauté the onions and garlic until soft and fragrant.

Add the flour to the skillet and stir until the onions and garlic are coated, then slowly add the chicken broth and milk, stirring constantly.

Once the mixture is smooth, add the frozen vegetables and bring to a simmer. Allow the mixture to thicken for about 5-10 minutes, stirring occasionally. If the mixture is too thick, add a bit more broth or milk.

Add the diced boiled chicken to the skillet and stir until everything is well combined. Season with salt and pepper to taste.

Pour the chicken and vegetable mixture into a pie dish.

Unroll the pie crust and place it over the top of the pie dish. Trim and crimp the edges, then make a few slits in the top of the crust to allow steam to escape.

Brush the top of the crust with the beaten egg.

Bake for 30-40 minutes, or until the crust is golden brown and the filling is bubbly.

Let the pie rest for 5-10 minutes before serving.