01 - Set the oven temperature to 350°F (175°C).
02 - Toss beef chunks with flour, salt, and pepper to coat evenly.
03 - Heat olive oil and butter in a large ovenproof pot over medium-high heat. Brown beef in batches, then remove and set aside.
04 - Add chopped onion and minced garlic to the pot; cook until softened, approximately 3 minutes.
05 - Incorporate mushrooms, carrots, and celery; sauté for 5 minutes, stirring occasionally.
06 - Stir in tomato paste, then return the browned beef to the pot.
07 - Pour in beef stock and red wine. Add Worcestershire sauce, thyme leaves, and bay leaf. Bring mixture to a simmer.
08 - Cover pot and transfer to the oven. Cook for 1.5 hours or until beef is tender.
09 - Remove pot from oven, discard bay leaf, and let filling cool briefly. Increase oven temperature to 400°F (200°C).
10 - Transfer filling to a deep pie dish. Roll out puff pastry to fit, cover filling, trim and crimp edges, then cut slits to vent steam.
11 - Brush the pastry surface with beaten egg evenly.
12 - Bake for 30 to 35 minutes until the pastry is golden and crisp.
13 - Allow the pie to rest for 10 minutes to settle before slicing.