In a large cast-iron or heavy skillet, on medium-high heat brown the lamb for 4 - 5 minutes then stir in the onion, garlic, celery, carrots, potato, salt, and pepper. Cook, stirring occasionally, 8 to 10 minutes.
Stir in the flour and cook, until incorporated, about 2 minutes. Pour in the stock and bring to a boil, then turn down to a simmer. Season the filling with salt and pepper to taste if needed.
Place the filling in an appropriately sized baking dish. Place the sheet of puff pastry over the top and trim the excess around the edges of the dish. Brush the puff pastry with some of the beaten egg, then make 2 - 3 small slices in the puff pastry to make a steam vent. Bake until the filling is bubbling and the pastry is golden about 30 minutes.