Preheat oven to 400°F (200°C) and grease a 9x13 baking dish with olive oil or cooking spray.
Place potato slices in a bowl, toss with olive oil, 1 teaspoon salt, and ½ teaspoon pepper.
Heat a large skillet over medium-high heat, add olive oil and ground meat, cook until browned and caramelized.
Add onions and carrots, cook until softened. Stir in garlic and cook for 30 seconds.
Mix in tomato paste, cook for 1-2 minutes. Add Worcestershire sauce and broth, scrape the bottom, and simmer with herbs for 5-7 minutes.
Remove from heat, stir in peas, and adjust seasoning.
Layer half the potatoes in the dish, add meat filling, then top with remaining potatoes. Brush with melted butter.
Cover with foil, bake for 30 minutes, uncover, then bake an additional 20-25 minutes until potatoes are golden.
Allow resting for 10 minutes before serving.