Place the peeled and quartered potatoes in a large pot and cover with cold water by about an inch. Add a generous pinch of salt. Bring to a boil, then reduce heat and simmer for 15-20 minutes, or until fork-tender.

Drain the potatoes well and return them to the hot pot over low heat for 1 minute to dry out any remaining moisture.

Remove from heat. Add the warmed milk, melted butter, salt, and pepper. Mash until smooth and creamy. Set aside.

Preheat oven to 375°F (190°C).

Heat olive oil in a large skillet or Dutch oven over medium-high heat. Add the ground lamb and cook, breaking it up with a spoon, until browned. Drain off any excess fat.

Add the chopped onion, carrots, and celery to the skillet. Cook for 5-7 minutes, or until vegetables begin to soften.

Stir in the minced garlic and cook for another minute until fragrant.

Sprinkle the flour over the lamb and vegetables, stir well, and cook for 1 minute, allowing the flour to coat everything.

Gradually pour in the beef broth, stirring constantly to prevent lumps. Add Worcestershire sauce, dried thyme, salt, and pepper. Bring to a simmer, then reduce heat to low and cook for 5-7 minutes, stirring occasionally, until the sauce thickens.

Stir in the frozen peas and corn. If using, stir in the fresh parsley. Remove from heat.

Pour the lamb filling evenly into a 9x13 inch baking dish.

Carefully spoon the mashed potato topping over the lamb filling, spreading it evenly to cover completely. You can use a fork to create decorative ridges on top for extra browning.

Place the baking dish on a baking sheet (to catch any potential overflow) and bake for 20-25 minutes, or until the filling is bubbly and the potato topping is golden brown.

If desired, you can place it under the broiler for the last 2-3 minutes to get a deeper golden crust, watching carefully to prevent burning.

Remove from oven and let stand for 5-10 minutes before serving. Garnish with fresh chopped parsley if desired.