Boil potatoes in salted water until fork-tender, about 18 minutes. Drain; steam-dry in empty pot a minute. Mash with butter and warm milk; season well.
Heat the oven to 400F. Brown the beef in an oven-safe skillet over medium-high heat, then spoon off the excess fat.
Add the oil, onion, and carrots and cook 6 minutes until softened. Add the garlic and tomato paste and cook 1 minute more until the paste darkens.
Stir flour into filling for 1 minute; add broth, Worcestershire, and thyme. Simmer 5 minutes until thick enough to hold a spoon trail. Stir in peas; season.
Spread the mash over the filling, rough up the surface with a fork, and bake 20 minutes. Broil 3 to 5 minutes for a browned top, then rest 10 minutes before serving.