Preheat the oven to 400°F (200°C).

Boil the chicken and vegetables in salted water until just cooked, about 15 minutes. Drain well.

Melt the butter in a skillet over medium-low heat. Add the flour and cook for 5 minutes to form a roux, whisking constantly. Gradually whisk in the milk or half-and-half and simmer until thickened, about 10 minutes. Dissolve the bouillon cubes in 1/4 cup warm water and add this to sauce along with Parmesan cheese. Stir until the cheese has melted.

Remove the skillet from the heat and fold in the boiled chicken and vegetables.

Transfer the mixture to a deep-dish pie plate and cover with the crust, sealing the edges well. Cut slits in the top for steam to escape and brush with beaten egg.

Bake for 45 to 50 minutes until the crust is golden and the filling is bubbling. Let rest for 10 to 15 minutes before serving.