In a saucepan, saute onion, garlic, and carrot, seasoned with salt and pepper. Once soft, add the sugar and cook for a minute before adding the tomatoes. Let cook while you get the polenta ready.
In an oven-proof skillet, boil the water. Slow pour in the polenta while stirring so it doesn’t clump. Cook for 3-4 minutes, and once thick add half the cheese and season with salt and pepper. Pour the sauce on top, and make 4 holes to crack in the 4 eggs. Bake for 7 minutes at 200C / 375F.