Preheat the oven to 425°F (220°C). Press one pie crust into the bottom and up the sides of a 9-inch pie dish.
In a large saucepan or Dutch oven, melt the butter over medium heat. Add the chopped onion and celery and cook until softened, about 5-7 minutes.
Stir in the flour, salt, pepper, and dried thyme. Cook for 1 minute, stirring constantly, to create a roux.
Gradually whisk in the chicken broth and heavy cream until smooth. Bring to a simmer and cook, stirring frequently, until thickened, about 5-8 minutes.
Remove from heat. Stir in the cooked chicken and thawed mixed vegetables until well combined. Pour the filling into the prepared pie crust.
Place the second pie crust over the filling. Trim excess dough, then pinch and crimp the edges to seal. Cut several small slits in the top crust for steam to escape.
Brush the top crust with the prepared egg wash. Bake for 30-35 minutes, until golden brown and bubbly. Place a baking sheet on the rack below to catch drips.
Let the pot pie cool on a wire rack for at least 10-15 minutes before slicing and serving. This allows the filling to set.