Place the potatoes in a medium pot and cover with water. Bring to a boil and cook until tender, about 12 minutes. Drain, then mash with 2 tablespoons butter, the milk, salt, and pepper until smooth. Set aside.
Heat the remaining 2 tablespoons butter in a large oven-safe skillet over medium heat. Add the ground beef and cook until browned, breaking it up with a spoon as it cooks.
Stir in the onion, carrots, and garlic. Cook for 5 minutes until the vegetables begin to soften.
Add the tomato paste, beef broth, Worcestershire sauce, and frozen peas. Simmer for 8 minutes until the mixture thickens slightly and the peas are heated through.
Remove the skillet from the heat. Spread the mashed potatoes evenly over the beef mixture, smoothing the top with the back of a spoon.
Place the skillet under the broiler for 3 to 4 minutes until the potato topping turns lightly golden. Watch closely to avoid burning.