01 - Preheat your oven to 375°F.
02 - Heat 1 tablespoon of oil in a large Dutch oven or heavy pot over medium-high heat. Brown the beef cubes in batches until nicely seared, then transfer the browned beef to a plate.
03 - Add the remaining oil, diced onion, sliced carrots, and sliced celery to the pot. Sauté for approximately 5 minutes, or until the vegetables have softened.
04 - Stir in the minced garlic and cook for 1 additional minute until fragrant.
05 - Return the browned beef to the pot. Sprinkle with flour and stir thoroughly to coat all the ingredients.
06 - Add the tomato paste, Worcestershire sauce, beef stock, non-alcoholic beer (or extra beef stock), dried thyme, bay leaves, salt, and black pepper. Stir well to combine all ingredients.
07 - Bring the mixture to a gentle 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 to a rich gravy. Remove and discard the bay leaves before proceeding.
08 - Transfer the prepared beef filling to a deep 9-inch pie dish, spreading it evenly.
09 - Roll out the thawed puff pastry to fit over the pie dish. Carefully drape the pastry over the filling, trim any excess dough, and crimp the edges to seal. Cut a few small slits in the top of the pastry to allow steam to escape during baking.
10 - Brush the surface of the puff pastry with the beaten egg.
11 - Bake in the preheated oven for 35 to 40 minutes, or until the pastry is puffed, golden brown, and crisp.
12 - Allow the pie to rest for 10 minutes after baking before slicing and serving.