Parboil the rice by combining it with 1/2 of the amount of boiling water that is required by the cooking directions. Cook, covered, over low heat, until all the liquid is absorbed.
The rice should be partially cooked, but still hard in the middle. Take off heat and let cool.
Combine beef, pork, parboiled rice, dill, parsley, and egg. Season with salt and ground black pepper. Mix well and taste for seasonings by microwaving or frying a small amount.
Cover the mixture and refrigerate for 1-2 hours.
Place tomatoes in a stock pot and cover with boiling water. Simmer, covered, for 20 minutes.
Drain and, using a wooden spoon, rub the tomatoes through a sieve. (Be careful! The tomatoes may squirt hot juice when broken up.)
Discard the skins and seeds remaining in the sieve. Cover the tomato sauce and set aside.
Warm the butter and 1/2 tbsp vegetable oil in a casserole, and saute onions and carrots, stirring from time to time, until golden, about 15 minutes.
Add garlic and saute, stirring, for 2 more minutes. Add tomato sauce, beef stock, and wine, and bring to a simmer.
Season with salt, black pepper, and cayenne. Add bay leaves, cover, and simmer over low heat for 15 minutes.
Warm some vegetable oil in a skillet or in a deep-fryer.
Shape the meat mixture into balls approximately 1-1/2 inches in diameter. Roll in flour and fry over high heat or deep-fry until a golden-brown crust is formed (the meatballs do not have to be completely cooked through at this stage).
Immerse the meatballs into the sauce, cover, and simmer over low heat for 1 hour.
Serve the meatballs in bowls or soup plates and spoon the sauce over them.