In a large frying pan, cook the onions & garlic in the butter & olive oil on medium heat for about 5 minutes. Stir in the tomatoes, ground beef, herbs, salt & pepper. Cook until the meat is browned thoroughly. Turn off heat & add egg yolk & Parmesan cheese, stirring to mix completely.
Spread the meat in the bottom of a lightly oiled oven proof dish (a 9x13 would be perfect). Spread the potatoes on top of the meat mixture. And finish by sprinkling the grated cheese on top.
Bake in 400 deg oven for 15 to 20 minutes, until the cheese is melted & the potatoes lightly browned.