Sauté the base: Heat olive oil in a skillet over medium heat. Cook onions, garlic, and carrots until softened.
Brown the lamb: Add ground lamb and cook until browned, breaking it apart as it cooks. Drain excess fat.
Build the flavor: Stir in tomato paste, Worcestershire sauce, thyme, salt, and pepper. Pour in broth and simmer for 10 minutes.
Add the peas: Stir in peas and cook for another 2 minutes. Remove from heat.
Prepare the potatoes: Boil potatoes until fork-tender. Drain and mash with butter, milk, salt, and pepper.
Assemble the pie: Spread the lamb mixture evenly in a baking dish. Top with mashed potatoes, smoothing them over with a spatula.
Bake: Bake at 375°F (190°C) for 30 minutes until golden brown.
Serve hot: Let it cool slightly before serving.