Preheat the oven to 425°F (220°C). In a large, deep skillet or Dutch oven, melt the butter with the olive oil over medium heat. Add the onion, celery, and carrots and cook for 8–10 minutes, stirring occasionally, until softened but not browned. Stir in the garlic and cook 1 minute more until fragrant.

Sprinkle the flour evenly over the vegetables and stir constantly for 1–2 minutes — the flour should coat everything and turn lightly golden. This cooks out the raw flour taste. Slowly pour in the warm chicken broth while stirring, then add the milk. Keep stirring as the sauce comes to a simmer and thickens, 4–5 minutes, until it coats the back of a spoon.

Stir in the poultry seasoning, salt, and pepper. Fold in the shredded rotisserie chicken, thawed frozen mixed vegetables, and parsley. Taste and adjust the seasoning — the filling should be boldly seasoned since the crust is plain. Let the filling cool for 10–15 minutes (a hot filling makes a soggy bottom crust).

Unroll one pie crust and line a 9-inch pie dish, letting the excess hang over the edge. Spoon in the cooled filling. Unroll the second crust and lay it over the top. Trim the overhang to about 1 inch, fold the edges under, and crimp with your fingers or a fork. Cut 4–5 slits in the top crust so steam can escape. Brush the entire top generously with the egg wash for a deep golden shine.

Place the pie on a rimmed baking sheet (to catch any drips) and bake at 425°F for 20 minutes. Reduce the oven to 375°F (190°C) and bake 25–30 minutes more, until the crust is deeply golden brown and the filling bubbles visibly through the vents. If the edges brown too fast, shield them with strips of foil.

Let the pot pie rest for 10–15 minutes before slicing. This is the hardest part — the rest lets the filling set so it slices cleanly instead of running everywhere. Serve hot.