Preheat the oven to 425°F (220°C).
In a large saucepan or Dutch oven, melt the butter over medium heat. Add the onions, carrots, and mushrooms, cooking for 8 to 10 minutes until the vegetables are softened and mushroom moisture has evaporated.
Whisk in the flour, salt, pepper, and thyme. Cook for 1-2 minutes to toast the flour without browning.
Gradually whisk in the chicken broth and heavy cream. Bring to a simmer and stir constantly until the sauce has thickened.
Add the cubed chicken and frozen peas into the sauce. Stir to combine and remove the pan from the heat.
Pour the chicken and mushroom mixture into a 9-inch deep-dish pie plate or similar capacity baking dish.
Place the pie crust over the filling. Trim, fold, and crimp the edges to seal them against the rim of the dish.
Cut four small slits in the center of the crust to allow steam to escape during the baking process.
Brush the entire surface of the crust with the beaten egg wash.
Bake for 30 to 35 minutes, or until the pastry is deep golden brown and the filling is bubbling through the vents.
Let the pot pie rest for 10 minutes at room temperature to allow the sauce to set before serving.