Prepare my pie crust recipe through step 5. Do this in advance to save time!

Heat the butter in a 10–12-inch oven-safe skillet over medium heat. Add the onion, garlic, carrots, and celery. Cook for 5 minutes or until vegetables have softened and released some liquid. Add flour, thyme, salt, and pepper. Stir and cook for 2 minutes as the flour soaks up the liquid.

Pour in the chicken broth and milk. Bring to a boil, reduce heat to low and allow to simmer for 5 minutes. Gravy will thicken. Stir in the vegetables and chicken/turkey. Cook for 1 more minute. Remove from heat and allow to slightly cool for 10 minutes (and up to 1 day—cover and store in the refrigerator).

Preheat oven to 375°F (191°C).

On a floured work surface, roll out one of the discs of chilled dough (you can freeze the 2nd for later use, see note). Turn the dough about a quarter turn after every few rolls until you have a circle 12 inches in diameter. If your skillet is 12 inches, roll out the pie dough to about 13 inches so it fits. Carefully lay the dough over the filling. Cut slits in the top to form steam vents. Crimp or flute the edges to seal. Lightly brush the top of the pie crust with the egg wash.

Bake for 35–40 minutes or until the top of the crust is golden brown. After the first 20 minutes of bake time, I place a pie crust shield on top of the skillet to prevent the edges of crust from burning.

Remove from the oven and allow to cool for 5 minutes before serving. Leftovers keep well in the refrigerator in an airtight container for up to 5 days. Reheat as desired.