Heat the oven to 400°F, then whisk the beaten egg with 1 tablespoon water and set the wash aside. The mixture should be loose enough to brush over the crust in a thin, even layer.

Melt the butter in a medium saucepan over medium-low heat and cook the diced onion for about 5 minutes until tender. The onion should soften and smell sweet without taking on much color.

Stir in the flour, poultry seasoning, thyme, salt, and pepper, then keep stirring for 2 minutes. The roux will look bubbly and pale golden and will smell lightly nutty.

Whisk in the chicken broth and milk or cream in gradual additions, stirring smooth after each one, then bring the sauce to a boil over medium heat and simmer for 1 minute. The finished gravy should thicken enough to coat the back of a wooden spoon without visible floury lumps.

Take the saucepan off the heat and fold in the cooked turkey, thawed vegetables, and cooked potato. Let the filling cool slightly so it is warm rather than steaming hot before it goes into the crust.

Fit one crust into a 9-inch pie plate, add the turkey filling, brush the rim with egg wash, and place the second crust over the top. Press and crimp the edges firmly so the filling stays contained as it bubbles.

Cut several slits in the top crust and brush it with the remaining egg wash. The vents give steam a path out, which helps prevent split edges and keeps the top pastry crisp.

Bake for 35 to 40 minutes. The crust should be lightly browned, with active bubbling visible through the center vents.

Rest the pie for 10 to 15 minutes before slicing. The gravy will settle and hold each slice together more cleanly.