Step 1

Filling: Preheat oven to 375 degrees. Heat a large skillet over medium-high. Swirl in oil. Add lamb; season with 1 teaspoon salt and cook, breaking up into bite-size pieces and stirring occasionally, until browned in places and just cooked through, 7 to 9 minutes. Transfer to a bowl.

Step 2

Remove all but 1 tablespoon fat from skillet; return to medium heat. Add onion and carrots; season with 1 teaspoon salt. Cook, stirring occasionally, until vegetables are softened and golden in places, 6 to 8 minutes. Stir in tomato paste; cook 30 seconds. Stir in flour; cook 1 minute. Return lamb and accumulated juices to skillet. Stir in Worcestershire, broth, and 1/4 teaspoon pepper. Bring to a boil, scraping up browned bits from bottom of skillet. Reduce heat to medium-low; simmer until thickened slightly, 1 to 2 minutes. Remove from heat. Stir in peas and parsley. Transfer to a 2-quart baking dish.

Step 3

Mash: In a large pot, cover potatoes with 1 inch of water; add 1 tablespoon salt. Bring to a boil, then reduce heat to medium and simmer until tender, 12 to 15 minutes; drain. Return pot to medium heat; swirl in oil. Add cabbage and 1/2 teaspoon salt. Cook, stirring occasionally, until collapsed and tender, 7 to 9 minutes. Return potatoes to pot; mash with a potato masher. Add milk and butter, mashing and stirring until mixture is creamy and cabbage is distributed evenly. Season with salt and pepper. Spread mash evenly over lamb mixture; brush top with butter. Place on a rimmed baking sheet lined with parchment.

Step 4

Bake until golden brown in places and bubbling along edges, 30 to 35 minutes. Let stand 10 minutes before serving. Pie can be refrigerated in an airtight container up to 2 days.