Heat oven to 375°. In a large pot, melt butter over medium. Add onion and celery and cook until onion is translucent, about 5 minutes. Add thyme and cook until fragrant, 1 minute. Add flour and cook, stirring constantly, until light golden, nutty and fragrant, about 2 minutes. Add half of broth, stirring constantly, until the mixture is smooth. Add the remaining broth and stir constantly until mixture comes to a rapid simmer and thickens, about 5 minutes. Stir in turkey and mixed vegetables and season with salt and pepper. Transfer to a 2-quart ovenproof baking dish. Set baking dish on a rimmed baking sheet lined with foil to catch any spills.
Place puff pastry on a lightly-floured work surface, dust with flour, and roll to a ⅛-inch thickness. Trim pastry so that it is 1-inch larger around than your baking dish. Using a sharp knife, slice a ¾-inch vent in the center of the pastry. Transfer the pastry to cover the pot pie filling, allowing the excess to hang over the sides of the dish. Lightly brush the pastry with the egg. Transfer to the oven and bake until the pastry lid is puffed and golden and the filling is bubbling in the center, about 40 minutes. Let cool 10 minutes before serving.
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