Place potatoes in large saucepan and add enough water to cover by 2 inches. Stir in 1 tablespoon salt and place pot over medium-high heat; bring to a boil. Cook until potatoes are fork tender, about 10 minutes. Turn off the heat and drain.

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Return potatoes to the hot pan and set over the same burner. Add milk and 2 tablespoons butter; allow butter to melt from the residual heat. Mash potatoes with a potato masher until smooth. Stir in Cheddar cheese, sour cream, egg yolk, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Set aside while you prepare the filling.

Preheat the oven to 375 degrees F (190 degrees C). Grease a 9×13-inch casserole dish.

Heat olive oil and remaining 1 tablespoon butter in a large skillet over medium-high heat. Add onion and carrot and cook, stirring constantly, until onion begins to turn translucent, about 5 minutes.

Add lamb and beef and cook, crumbling with a spoon and stirring often until browned, about 5 minutes. Sprinkle meat mixture with flour, remaining salt and pepper, rosemary, and thyme. Cook, stirring constantly, for 2 minutes.

Add beef stock, tomato paste, and Worcestershire; cook, scraping any flavorful bits from the bottom of the skillet. Bring mixture to a simmer. Cook until thickened and vegetables are tender, about 8 minutes. Stir in peas. Transfer mixture to the prepared casserole dish and top with the prepared mashed potatoes.

Bake in the preheated oven until golden and bubbly around the edges, 25 to 30 minutes. Sprinkle with parsley and serve.