01 - Preheat your oven to 375°F. In a large Dutch oven or heavy pot, heat 1 tablespoon of vegetable oil over medium-high heat. Brown the beef cubes in batches until seared on all sides, then remove and set aside on a plate.
02 - Add the remaining 1 tablespoon of oil, diced onion, sliced carrots, and sliced celery to the pot. Sauté for approximately 5 minutes, or until the vegetables begin to soften.
03 - Incorporate the minced garlic and cook for an additional minute until fragrant. Return the browned beef to the pot. Sprinkle with all-purpose flour and stir thoroughly to coat the beef and vegetables.
04 - Stir in the tomato paste, Worcestershire sauce, beef stock, non-alcoholic dark beer (or additional beef stock), dried thyme, bay leaves, salt, and black pepper, ensuring all ingredients are well combined.
05 - Bring the mixture to a gentle simmer, then cover the pot and reduce heat to low. Cook for 1 hour and 30 minutes, stirring occasionally, until the beef is fork-tender and the sauce has thickened to a rich consistency. Remove and discard the bay leaves.
06 - Carefully spoon the prepared beef filling into a deep 9-inch pie dish.
07 - Roll out the thawed puff pastry sheet to fit the pie dish. Drape the pastry over the beef filling, trim any excess dough, and crimp the edges to seal. Cut several small slits into the pastry top to allow steam to vent during baking.
08 - Brush the surface of the puff pastry with the beaten egg wash for a golden finish.
09 - Bake for 35 to 40 minutes, or until the pastry is beautifully golden brown and puffed.
10 - Allow the pie to rest for 10 minutes after baking before slicing and serving.