Boil the potatoes in salted water until a fork slides in easily, about 18 minutes. Drain them and let them steam dry in the empty pot for a minute, then mash with the butter and warm milk and 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.
Sprinkle the flour over everything and stir for 1 minute, then pour in the broth and Worcestershire. Add the thyme and simmer 5 minutes until thick enough to hold a spoon trail. Stir in the peas and 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.