In a large bowl, blend the ground beef, eggs, breadcrumbs, and grated carrots. Heat olive oil in a large skillet on medium heat. Form meat mixture into 1 inch balls and brown in skillet on all sides. Work in batches, setting aside browed meatballs, removing excess fat from skillet as needed.

In a large pot, add the tomato puree, tomato sauce, cranberry sauce, apples, sugar, lemon juice, water and onion. Combine well and simmer over low heat for about 15-20 minutes, stirring frequently to keep sauce from burning on the bottom.

Transfer sauce to a stock pot. Add the meatballs to the stock pot, stirring carefully just to cover the meatballs with sauce. Cover, and simmer over low heat for 45 minutes, stirring occassionally. Add water, if necessary, to thin out the sauce. Serve over a bed of rice.