Make pie crust and place in fridge to chill while preparing the filling.
1 unbaked pie crust
Preheat oven to 400°F.
In a large pot or Dutch oven, melt butter over medium heat. Add the celery, carrots, onion, and peas and cook, stirring frequently, until the vegetables start to soften and turn translucent, about 4-5 minutes.
4 Tablespoons salted butter, 3 celery stalks, 3 medium carrots, 1 large yellow onion, 1/2 cup frozen peas
Stir in the chopped chicken, then sprinkle the flour evenly over the filling mixture and stir to combine. Cook and stir for 2 minutes.
2 cups cooked chicken, 1/4 cup all-purpose flour
Pour the chicken broth over the chicken and vegetable filling mixture, stirring constantly so there are no lumps of flour remaining. Stir in chicken bouillon.
2 cups chicken broth, 1 teaspoon chicken bouillon
Add the cream and stir to combine. Reduce heat to low and cook for 4-5 minutes until thickened slightly.
1 cup heavy cream
Add thyme, salt, and pepper and stir to combine. Taste and adjust seasoning as needed, then pour filling into a deep pie plate or casserole dish.
1 teaspoon ground thyme, 1 teaspoon kosher salt, 1/4 teaspoon freshly ground black pepper
Roll out crust on a lightly floured surface and gently lay on top of the filling. Cut small slits in the top for the pie to vent while it bakes. Press the crust into the sides of the dish to seal, then brush with the egg wash.
1 large egg
Bake for 30-40 minutes until the crust is golden brown and the filling is hot and bubbly. Cool for 10-15 minutes before serving.