Heat the oven to 425°F. Place one pie crust in a 9-inch deep-dish pie plate and keep the second crust chilled until the filling is ready.

Melt the butter in a large skillet or saucepan over medium heat. Add the onion, carrots, and celery, then cook for 7 to 9 minutes, stirring occasionally, until the vegetables begin to soften.

Sprinkle the flour over the vegetables and stir continuously for about 1 minute. This cooks the flour and helps prevent a raw flour flavor in the finished sauce.

Gradually whisk in the stock, followed by the milk. Cook for 3 to 5 minutes, whisking as needed, until the sauce reaches a gentle boil and thickens.

Stir in the salt, pepper, thyme, optional sage, turkey, peas, and parsley. Remove the pan from the heat, taste for seasoning, and let the filling cool for about 10 minutes.

Spoon the cooled filling into the bottom crust. Cover with the top crust, trim the edges, and crimp or press the crusts together to seal.

Cut 4 to 6 small vents in the top crust. Beat the egg with 1 tablespoon water, then brush the mixture evenly over the pastry.

Bake at 425°F for 15 minutes. Reduce the oven temperature to 375°F and continue baking for 30 to 40 minutes, until the crust is deeply golden and the filling bubbles through the vents.

If the crust browns too quickly, loosely cover the edges or the entire pie with aluminum foil. Let the finished pie rest for at least 10 to 15 minutes before slicing.