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.

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 9x13-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.