In a large pot or Dutch oven, brown the ground lamb or beef over medium heat until fully cooked. Drain excess fat if necessary.
Add the diced onion, carrots, and celery to the pot. Sauté for about 5 minutes, or until the vegetables soften.
Stir in the minced garlic, and cook for an additional 1-2 minutes until fragrant.
Add the frozen peas, beef broth, tomato paste, Worcestershire sauce, thyme, rosemary, salt, and pepper. Bring to a boil, then reduce heat and let simmer for 25 minutes.
While the soup is simmering, boil the cubed potatoes in a separate pot of salted water until fork-tender, about 15 minutes. Drain the potatoes.
In a mixing bowl, mash the drained potatoes with milk, butter, and a pinch of salt until smooth and creamy.
Once the soup is done, taste and adjust seasoning as needed. Serve the hot soup in bowls topped with a generous scoop of the creamy mashed potatoes.