Boil the potatoes in a large pot of salted water until tender, about 15-20 minutes. Drain well.
Mash the potatoes with the warmed milk, butter, salt, and pepper until smooth and creamy. Set aside.
In a large skillet or Dutch oven, brown the ground lamb over medium-high heat. Drain off any excess fat.
Add the chopped onion, diced carrots, and diced celery to the skillet. Cook until softened, about 5-7 minutes.
Stir in the minced garlic and cook for 1 minute more until fragrant.
Sprinkle the flour over the meat and vegetable mixture and stir to coat. Cook for 1 minute.
Gradually stir in the beef broth and Worcestershire sauce. Bring to a simmer, then reduce heat and cook for 10 minutes, stirring occasionally, until the sauce thickens.
Stir in the dried thyme, salt, pepper, and frozen peas. Cook for 2 minutes.
Pour the lamb filling into a 9x13 inch baking dish.
Spread the mashed potatoes evenly over the lamb filling.
Bake for 20-25 minutes, or until the potato topping is golden brown and the filling is bubbly.
Let stand for 10 minutes before serving.
Notes
This Shepherd's Pie is best served hot. You can also make it ahead of time and reheat it in the oven.