Brown the Beef: Heat vegetable oil in a large Dutch oven over medium-high heat. Season the beef cubes with salt and pepper, then brown them in batches until nicely seared on all sides.

Sauté Vegetables: Add the diced onion, carrots, and celery to the pot. Cook for 5-7 minutes until the vegetables begin to soften. Add the minced garlic and cook for another minute until fragrant.

Create the Gravy: Sprinkle the flour over the vegetables and beef, stirring constantly for 2 minutes to cook out the raw flour taste. Gradually pour in the beef broth while stirring to prevent lumps. Add the thyme and Worcestershire sauce.

Simmer and Thicken: Bring the mixture to a simmer, then reduce heat to low. Cover and cook for 1.5-2 hours until the beef is fork-tender. Stir in the frozen peas during the last 5 minutes of cooking.

Assemble the Pie: Preheat your oven to 400°F (200°C). Transfer the beef filling to a 9-inch pie dish. Roll out one sheet of pastry to cover the bottom if using a deep dish, then place the second sheet over the top. Crimp the edges and cut slits in the top for steam to escape. Brush with beaten egg for a golden finish.

Bake to Perfection: Bake for 30-35 minutes until the crust is golden brown and the filling is bubbling. Let it rest for 10 minutes before serving to allow the filling to set.