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Divide the meat mixture into tablespoons and roll into smooth balls in your hands. Set rolled meatballs aside on a plate or other flat surface.
In a 4 1/2 quart soup pot or large saucepan, heat the olive oil over medium high heat until hot. Add half of the rolled meatballs and cook, stirring occasionally, until browned on all sides, about 4 minutes. Transfer the browned meatballs to a plate and set aside. Brown the remaining meatballs in the same manner and set aside.
Add the chopped onion and celery to the pot and cook, stirring, until vegetables are soft, 3 to 4 minutes. Add the remaining teaspoon of minced garlic and cook for 1 minute. Add the tomato paste, tomatoes and their juices, beef broth, 3 cups water, remaining 1/4 teaspoon of Italian Essence, remaining 1/2 teaspoon of salt, remaining pinch of crushed red pepper and stir to combine. Return the meatballs to the soup pot and bring the soup to a boil. Reduce the heat so that the soup just simmers and cook for 30 minutes.
Using a spoon, carefully skim any fat that has accumulated on the top of the soup and discard. Add the ditalini to the hot soup, stir well, and cook for 15 minutes, or until the pasta is cooked through. Stir in the chopped basil and serve the soup in wide bowls, garnished with additional grated Parmesan, if desired.