Boil potatoes with salt until tender (8–10 min); mash with milk, egg, butter, salt, and pepper.
Sauté carrots, onion, and salt in butter (5 min); add beef and cook until no pink remains (8–10 min).
Stir in tomato paste, garlic, and thyme (1 min); sprinkle flour and cook (1 min).
Add broth and Worcestershire; simmer until thickened (10 min); stir in peas.
Pipe mash over filling; broil until golden (5–10 min); cool 10 min before serving.