Preheat the oven to 425°F.
Heat 1 tablespoon of oil in a large oven-proof skillet over medium-high heat. Add the ground beef and cook until no longer pink, breaking up the meat as you stir, about 5-7 minutes. Drain off the fat and transfer the beef to a separate bowl.
Heat additional 1 tablespoon of oil in the now-empty skillet over medium heat until shimmering. Add mushrooms, carrots, and onion and cook over medium-high heat until softened and slightly browned, about 5-7 minutes. Stir in tomato paste and garlic and cook until fragrant, about 30 seconds. Stir in flour and cook for 1 minute.
Stir in wine and cook until almost completely evaporated, about 2 minutes. Gradually add the broth and Worcestershire sauce, stirring and scraping any bits from the bottom of the pan. Bring to a boil, then reduce the heat and simmer gently until the mixture thickens slightly, about 3-4 minutes. Stir in the ground beef, frozen peas, and thyme. Taste the filling and season with salt and pepper, as necessary.