Mix beef, onion, mint, parsley, salt, pepper, and rice. Add 1/4 cup beef stock, and mix ingredients together. Form into walnut-sized meat balls.
In a large pot, bring the remaining stock and water to a boil. Drop the meatballs into the boiling liquid, and lower the heat. Simmer on low for 45 minutes.
In a bowl, beat the egg yolks and add the lemon juice. While beating, add a few spoons of the hot stock into the yolks to temper them. Then add the yolk mixture to the pot of meatballs, and stir constantly until incorporated. Cover the pot and remove from the heat. Let stand 5 minutes before serving.