Peel and chop the onion and set it aside. Rinse the celery, trim off the ends, and dice the celery. Set it aside with the onions. Rinse the carrots, peel them, trim off the ends, and slice the carrots into coins about ⅓-inch thick.
In a tall (8-quart) soup pot, heat the olive oil on medium-high. Add the onions, celery and carrots, and stir to combine. Let the veggies cook for five minutes, or until the onions are starting to soften, stirring occasionally.
Add the frozen cauliflower and green beans to the pot, give them a stir, and cook for another five minutes. Add the beans and fire-roasted tomatoes, then the stock, bay leaf, seasonings and Liquid Smoke. (The beans added early will cook down until almost invisible, so if you prefer them intact, add the beans to the soup later, during the last several minutes of cooking.)
Cut off the rind from a triangle of Romano cheese and add it to the pot. Stir the soup and let it cook on medium high, covered, until it boils, then lower it slightly. Let the soup cook for 20-30 minutes, with the lid propped open by a wooden spoon, until the vegetables are tender.
As soon as you add in the Romano rind, you can start getting your pasta ready. Fill a pasta pot just over half-way full with hot water, add salt, and cover the pot. Heat it on high until it comes to a boil. Stir in the pasta and cook, uncovered, until it's al dente, stirring occasionally to prevent sticking. When the pasta is done, drain it in a colander in the sink and return the pasta to your cooking pot. Ladle some soup broth into your pasta to prevent it from sticking. Turn off the soup when it's done.
To serve the minestrone, ladle some pasta into each bowl and add the soup over it. Sprinkle generously with grated Romano or Parmesan.
If you used chicken stock, store the soup for up to four days in the refrigerator (separately from the pasta). If you used vegetable stock, the soup can stay for a week. You also could freeze the minestrone without the pasta and use it within three months for best quality.
To Make a Smaller BatchIf you want to make roughly half a batch, use one quart of stock. Use half of the carrots and frozen vegetables and make half a pound of pasta.Keep the same amount of tomatoes and rind of cheese. For the seasonings and Liquid Smoke, start with half the amount and taste to see if more is needed.Making Minestrone with MeatFor chicken minestrone, stir in cooked, shredded chicken during the last few minutes of cooking, as in this Chicken Pastina Soup.You also could make minestrone with sausage by browning some ground Italian sausage in the pot when the carrots, celery and onion are soft. For beef minestrone soup, use ground sirloin.