01 - Set oven to 350°F.
02 - Toss beef cubes with flour, salt, and pepper.
03 - Heat olive oil in large heavy-bottomed pot over medium-high heat. Brown beef in batches; set aside.
04 - In the same pot, sauté chopped onion for 4 minutes until softened, then add garlic and cook for 1 minute.
05 - Incorporate mushrooms and carrots, cooking until mushrooms release liquid and begin to brown, about 5 minutes.
06 - Stir in tomato paste and cook for 1 minute, then return beef to the pot.
07 - Pour in beef broth and red wine. Stir in Worcestershire sauce, thyme, and bay leaf. Bring to simmer.
08 - Cover and transfer pot to oven. Braise for 1½ hours or until beef is tender.
09 - Remove bay leaf. Adjust seasoning and allow filling to cool to room temperature.
10 - Raise oven to 400°F.
11 - Roll out puff pastry to fit a 9-inch pie dish.
12 - Spoon cooled filling into dish, cover with pastry, trim and crimp edges. Cut slits for steam.
13 - Brush pastry surface with beaten egg.
14 - Bake for 30 to 35 minutes until pastry is puffed and golden.
15 - Allow pie to rest for 10 minutes before slicing.