Heat the oven. Set a rack in the center and heat the oven to 400°F. Take the puff pastry out of the freezer so it can thaw while the filling cooks — no more than 40 minutes at room temperature, or the butter melts into the dough and the layers will not puff.

Soften the aromatics. Melt the butter in a large Dutch oven or deep skillet over medium heat. Add the onion, carrot, celery, garlic, thyme, and rosemary, and cook until the onion turns translucent, about 3 minutes.

Add the mushrooms and potatoes. Stir in the mushrooms and cook 1 minute, then add the potatoes, salt, and pepper. Cook 6 to 7 minutes, stirring now and then, until the potatoes start to soften at the edges.

Build the gravy. Sprinkle the flour over the vegetables and stir for about 2 minutes, until the raw flour smell fades. Pour in the chicken stock and milk a little at a time, stirring constantly, until the sauce turns smooth. Let it simmer 5 to 6 minutes, until slightly thickened and the potatoes are tender but still holding their shape.

Fold in the turkey and peas. Stir in the turkey and peas, turn off the heat, and taste. Add salt and pepper until it tastes right.

Fill the dish. Spoon the filling into a 9-inch deep-dish pie plate or round baking dish. Roll the puff pastry out on a lightly floured surface until it covers the dish with a little overhang. Lay it over the filling, trim the edges, and press to seal.

Egg wash and vents. Whisk the egg with 1 tablespoon of milk and brush it over the pastry. Cut 4 small slits in the top, about 1/2 inch long, so steam can escape.

Bake. Set the dish on a baking sheet and bake until the pastry is puffed and deep golden brown, 45 to 55 minutes. Let it rest about 10 minutes before serving so the filling settles.