Bring a large pot of salted water to a boil. Add potatoes and cook until tender but still firm, about 15 minutes. Drain and mash. Mix in butter, finely chopped onion, and 1/4 cup shredded cheese. Season with salt and pepper to taste; set aside.

Bring a large pot of salted water to a boil. Add carrots and cook until tender but still firm, about 15 minutes. Drain, mash, and set aside.

Preheat oven to 375 degrees F (190 degrees C).

Heat oil in a large frying pan. Add onion and cook until clear. Add ground beef or lamb and cook until well browned. Pour off excess fat, then stir in flour and cook 1 minute. Add ketchup and beef broth. Bring to a boil, reduce heat, and simmer for 5 minutes.

Spread the ground beef or lamb in an even layer on the bottom of a 2-quart casserole dish. Next, spread a layer of mashed carrots, then top with the mashed potato mixture, and sprinkle with remaining shredded cheese.