01 - Preheat the oven to 400°F.
02 - Season beef cubes generously with salt and pepper. Heat olive oil in a large Dutch oven or heavy pot over medium-high heat. Brown beef in batches without overcrowding, transferring to a plate as each batch browns.
03 - Reduce heat to medium. Add onion, garlic, carrots, celery, and mushrooms to the pot. Sauté until vegetables are softened, about 7 minutes.
04 - Stir in tomato paste and flour. Cook for 1 minute, stirring constantly to incorporate and remove raw flour taste.
05 - Return browned beef to the pot. Add beef broth, red wine, Worcestershire sauce, thyme, rosemary, and bay leaf. Stir well to combine and bring to a gentle simmer.
06 - Cover and cook on low heat for 1 hour, stirring occasionally, until beef is tender and sauce has thickened. Remove and discard bay leaf.
07 - Stir in frozen peas. Taste and adjust salt and pepper if needed.
08 - Transfer the beef filling to a 9-inch deep-dish pie dish or ovenproof casserole.
09 - Roll puff pastry to fit over the dish. Place pastry over filling, trim excess edges, and press firmly to seal. Cut 4-5 small slits in the top to vent steam during baking.
10 - Brush pastry surface evenly with beaten egg.
11 - Bake for 25-30 minutes until pastry is golden brown and filling is bubbling through vents.
12 - Allow pot pie to cool for 10 minutes before serving to set filling and prevent burning.