In a large stock pot, bring water to a boil; stir in salt. Add potatoes and boil gently for about 15 to 20 minutes or until they are soft. Drain off water and add butter and salt and pepper. Mash the potatoes, adding enough milk to make a smooth mixture. Set aside.
Preheat the oven to 350 degrees. In a large skillet, heat the oil over medium-high heat for 1 minute. Add the onion and stir-fry until tender. Stir in ground beef; then add carrot, parsley, garlic, and thyme. Cook until lamb is thoroughly browned, stirring frequently. Add Worcestershire sauce and tomato paste- stir well. Spread the meat mixture evenly in the bottom of a deep pie dish.
Top with mashed potatoes and cheese.
Bake in preheated oven for 20 minutes covered with Foil, remove foil and bake for another 10 minutes.
USDA recommends ground lamb reach an internal temp of 160F