Brown the Beef: In a heavy pot, heat olive oil over medium-high heat. Add the beef chuck in batches, browning for 5-7 minutes. Remove and set aside.

Sauté the Vegetables: In the same pot, add more olive oil if needed, sauté onion for 4-5 minutes, then add garlic, carrots, and mushrooms and cook for 5-6 minutes.

Create the Gravy Base: Stir in tomato paste and flour, cooking for 1-2 minutes until well combined.

Deglaze the Pot: Pour in red wine, scraping up bits from the bottom. Let simmer for 2-3 minutes, then add beef stock, beef, thyme, bay leaves, salt, and pepper.

Simmer the Filling: Cover and simmer for 1.5 to 2 hours, stirring occasionally until beef is tender.

Prepare the Puff Pastry: Preheat oven to 400°F (200°C). Roll out puff pastry to cover pie dish.

Assemble the Pie: Transfer filling into a pie dish, drape puff pastry over, trim edges, and cut a steam slit.

Bake the Pie: Brush pastry with beaten egg and bake for 25-30 minutes until golden.

Let it Rest: Remove pie from oven and let rest for 5-10 minutes before serving.