01 - Preheat oven to 350°F. Toss beef cubes with flour, salt, and pepper.
02 - Heat olive oil and butter in a large ovenproof pot over medium-high heat. Brown beef in batches and set aside.
03 - Add onion and garlic to pot; cook until softened, about 3 minutes.
04 - Add mushrooms, carrots, and celery; sauté for 5 minutes, stirring occasionally.
05 - Stir in tomato paste, return beef to pot, then add beef stock, red wine, Worcestershire sauce, thyme, and bay leaf. Bring to a simmer.
06 - Cover pot and transfer to oven. Braise for 1.5 hours until beef is tender.
07 - Remove pot from oven, discard bay leaf, and let filling cool slightly. Increase oven temperature to 400°F.
08 - Spoon filling into a deep pie dish. Roll out puff pastry to fit, place over filling, trim and crimp edges, then cut slits for steam release.
09 - Brush pastry surface with beaten egg.
10 - Bake for 30 to 35 minutes until pastry is golden and crisp.
11 - Allow pie to rest for 10 minutes before serving.