Season the beef cubes evenly with salt and pepper. Heat the vegetable oil in a large Dutch oven over medium-high heat until shimmering, then brown the meat in batches until a crust forms on all sides, about 5 minutes per batch.
Add the onion, carrots, and celery to the pot and cook until the vegetables soften and the onion turns translucent, about 8 minutes. Stir in the garlic and cook until fragrant, about 1 minute.
Sprinkle the flour over the mixture and stir constantly for 1 minute to coat everything. Gradually pour in the beef broth while stirring to prevent lumps, then add the Worcestershire sauce, tomato paste, and thyme.
Bring the mixture to a simmer, then reduce heat to low, cover, and cook for 1 hour and 30 minutes until the beef is fork-tender and the gravy has thickened. Stir in the frozen peas and remove from heat.
Preheat your oven to 375°F. Transfer the filling to a 9-inch pie dish and place the pie crust over the top, crimping the edges against the dish. Cut 3-4 small slits in the center of the crust to vent steam, then brush the entire surface with the egg wash.
Bake for 30-35 minutes until the crust is golden brown and the filling is bubbling. Let the pie rest for 10 minutes before serving to allow the filling to set.