Preheat the oven to 400 degrees F. In a large skillet over medium heat, cook the ground beef until it is browned and no longer pink. Drain any excess fat.
Add the diced onion and carrots to the skillet. Cook for 5 minutes until the vegetables begin to soften.
Sprinkle the flour over the beef mixture and stir well to coat everything evenly. Cook for 1 minute.
Slowly pour in the beef broth while stirring. Add the Worcestershire sauce, dried thyme, salt, and pepper. Simmer until the mixture thickens into a gravy, about 3 minutes.
Stir in the frozen peas and remove the skillet from the heat.
Fit one pie crust into a 9-inch pie dish. Pour the beef filling into the crust. Place the second crust on top, crimp the edges to seal, and cut a few small slits in the center. Brush the top with melted butter.
Bake for 25 to 30 minutes until the crust is golden brown. Let the pie rest for 5 minutes before slicing.