Preheat the oven to 425ºF.

Place the potatoes and the carrots in a pot and cover with water. Bring to a boil, reduce the heat and cook until the vegetables are just tender, 8-10 minutes. Drain.

In a large skillet, heat the olive oil. Add the mushrooms and cook until they start to soften, then add the onions. Cook until the mushrooms and onions are both softened, then remove to a bowl.

In the same skillet, melt the butter. Whisk in the flour and cook for a minute. Stir in the salt, thyme and pepper. Slowly whisk in the chicken broth and the milk. Bring the mixture to a boil then stir until thickened.

Remove from the heat and stir in the chicken, peas, potatoes and carrots, and mushrooms and onions.

Roll out 1 of the crusts and place in a  9-inch pie plate. Fill with the chicken mixture. Roll out the remaining pie crust and place over the filling. Trim and seal the edges. Cut slits in the top the crust.

In a small bowl, whisk the egg with about a tablespoon of water. Brush the egg wash over the crust of the pie. Set the dish on a baking sheet (to catch anything that may bubble over while baking).

Bake the pie until the crust is golden, 30-40 minutes. Watch it to make sure it doesn’t over-brown.

Let the pie stand for 15 minutes before slicing.