Preheat the oven to 375°F (190°C).
In a large skillet, heat a tablespoon of oil over medium heat. Add the diced onion and cook until translucent, about 5 minutes.
Add the minced garlic and sauté for an additional minute until fragrant.
Stir in the ground beef, breaking it up with a spoon. Cook until browned, about 8-10 minutes.
Sprinkle the flour over the mixture, stirring to combine, and cook for another 2 minutes.
Pour in the beef broth, Worcestershire sauce, and add the thyme, salt, and pepper. Stir well to combine.
Add the diced carrots and frozen peas, stirring to ensure even distribution. Let simmer for 5 minutes over low heat.
Remove from heat and let the filling cool slightly for easier handling.
Roll out one pie crust and fit it into a pie dish. Pour the meat mixture into the crust and spread evenly.
Cover with the second pie crust, sealing the edges by crimping them together. Cut slits on top for steam to escape.
Brush the top of the pie with beaten egg for a golden finish.
Bake in the preheated oven for 45 to 50 minutes, or until the crust is golden brown.
Let the pie rest for 10 minutes before slicing and serving.