Heat up 3 tablespoons of olive oil in a large stockpot or Dutch Oven over medium heat. Add in your onions, carrots, and celery and saute until soft (about 5 minutes). Add and garlic and stir until fragrant, about 30 seconds.
Add in the crushed tomatoes, potatoes, zucchini, cabbage, basil, beans, and tomato paste until combined. If using fresh or frozen green beans, add them in now as well.
Add in vegetable broth and enough water to cover the ingredients by about an inch.
Raise the heat to high until the soup comes to a boil. Reduce heat to low and simmer for about and 1.5 hours or until beans are tender.
Add in the 2/3 cup pasta, salt, pepper, balsamic vinegar, and cook for another 10ish minutes until pasta is cooked.
Once the soup is done, taste to be sure you have enough salt, pepper, and acid. Adjust accordingly.
To serve, drizzle 1/2-1 tablespoon of olive oil into a bowl, add the minestrone and enjoy! If you choose, top with a sprinkle of parmesan and a piece of toasted bread.