Preheat your oven to 375°F (190°C).
In a large skillet over medium heat, add the ground lamb or beef and cook until browned. Drain any excess fat if necessary.
Add the chopped onion, diced carrots, and minced garlic to the skillet. Cook for about 5 minutes, until the vegetables soften.
Stir in the tomato paste, Worcestershire sauce, dried thyme, salt, and pepper. Pour in the beef or vegetable broth, and let the mixture simmer for 10 minutes, allowing the flavors to meld.
Add the peas to the skillet, stirring them into the meat and vegetable mixture. Simmer for an additional 5 minutes, ensuring everything is well combined.
While the filling simmers, place the potato chunks in a large pot of salted boiling water. Cook for about 15 minutes or until the potatoes are tender when pierced with a fork.
Drain the potatoes and return them to the pot. Add the butter, milk, salt, and pepper, then mash the potatoes until smooth and creamy.
In a greased baking dish, spread the meat and vegetable filling evenly. Top it with the creamy mashed potatoes, using a fork to create decorative lines on the surface.
Bake the Shepherd's Pie in the preheated oven for 20-25 minutes, or until the mashed potatoes turn lightly golden and slightly crispy.