01 - Preheat the oven to 375°F (190°C).
02 - Heat 1 tablespoon of vegetable oil in a large Dutch oven or heavy pot over medium-high heat. Brown the beef cubes in batches, transferring them to a plate once done.
03 - Add the remaining oil, diced onion, sliced carrots, and sliced celery to the pot. Sauté for approximately 5 minutes until they begin to soften.
04 - Stir in the minced garlic and cook for an additional 1 minute until fragrant.
05 - Return the browned beef to the pot. Sprinkle with all-purpose flour and stir thoroughly to ensure the beef is well coated.
06 - Stir in the tomato paste, Worcestershire sauce, beef stock, non-alcoholic dark beer (or extra beef stock), dried thyme, bay leaves, salt, and black pepper. Mix to combine all ingredients.
07 - Bring the mixture to a simmer, then cover the pot and reduce heat to low. Cook for 1.5 hours, stirring occasionally, until the beef is tender and the sauce has thickened. Remove and discard the bay leaves.
08 - Transfer the prepared beef filling into a deep 9-inch pie dish.
09 - Roll out the thawed puff pastry to an appropriate size to fit over the pie dish. Drape the pastry over the filling, trim any excess dough, and crimp the edges securely. Cut a few small slits into the pastry to allow steam to escape during baking.
10 - Brush the top surface of the puff pastry evenly with the beaten egg.
11 - Bake in the preheated oven for 35 to 40 minutes, or until the puff pastry is golden brown and crisp.
12 - Allow the baked pie to rest for 10 minutes before slicing and serving.