Preheat the oven to 400 degrees F.
Carefully unfold one sheet of puff pastry on a floured surface. Using a rolling pin, roll out the pastry large enough to cover the bottom of 9-inch pie plate with overhang (about 12 inches in diameter). Trim the overhang, and using the tongs of a fork, prick the bottom of the pastry all over.
In a saucepan over medium-high heat, melt the butter. Add the carrots, celery and onions and cook until the vegetables start to soften, about 5 minutes. Stir in the flour and cook for 2 minutes, stirring constantly. Slowly add the broth and milk. Bring to a boil, reduce to a simmer and cook until the mixture thickens enough to coat the back of a spoon.
Stir in the turkey, peas, celery seed and season with salt and pepper. Remove from the heat and cool completely.
Unfold an additional sheet of pastry on a floured surface and cut into 6 to 8 even strips.
Pour the filling into the pie crust. Create a lattice pattern over the top by weaving the strips over and under each other. Trim the excess edges. Gently beat the egg and 1 tablespoon water in a small bowl. Brush the dough with the egg wash before placing the pot pie on a sheet tray.
Place in the oven and bake until the pastry is golden brown and the filling is bubbly, 45 minutes. Cool for 10 minutes before serving.