In a tall (8-quart) stock pot over medium high heat, add the beef chuck, stock, and hot water, making sure you don't fill your pot more than ¾ full. (When the liquid starts to boil, lower the heat to a simmer.) During the first 20 minutes of cooking, use a slotted spoon or fine mesh strainer to skim off the foam that comes to the surface and discard it.
While the beef is cooking, prepare the vegetables. Peel and quarter the onion. Wash the celery, trim off the ends, and cut into 1-inch pieces. Rinse the carrots and trim off the ends. Peel the carrots and cut them in half horizontally. Rinse the parsley and remove the stems (with scissors, if you have them). Peel and chop the garlic. Rinse the basil and remove the leaves.
After 20 minutes, add the onion, garlic, tomatoes, vegetables, herbs and seasonings to the pot and give it a quick stir. Cover the pot and keep the heat on low. Cook the soup for four hours, but remove about half of the carrots after 35 minutes. (You can refrigerate them after they cool and start bringing them to room temperature an hour before serving the soup.)
When the soup is done, remove the meat to a bowl and either shred it or cut it into chunks. Strain the soup into another pot (place a strainer over the empty pot and pour the soup through the strainer to remove the vegetables.) Discard the peppercorns. The vegetables in the strainer will be mushy, but you can save them in the refrigerator to eat later with salt and pepper if you wish.
Taste the soup to see if it needs additional salt and pepper.
For the pasta, fill a pot ⅔ full with hot water, add a little salt. Cover the pot and heat on high until rapidly boiling. Add the pasta and cook it until it's al dente, stirring occasionally so it doesn't stick. Turn off the heat, drain the pasta and return it to the pot. Scoop a little soup broth into the pasta so it doesn't stick.
To serve, scoop some pasta and sliced, reserved carrots into each bowl and add some shredded beef, if desired. Ladle the broth over it. Top each serving with freshly grated Parmesan and have salt and red pepper flakes at the table for extra seasoning. If you prefer, the beef can be served separately as a second dish.
To store leftover pasta, remove it from the pot with a slotted spoon so there is no liquid in it to make it soggy. Store it separately from the soup in the refrigerator. The leftover soup and pasta should be good for four days. If you wish to freeze leftover soup, place it in a plastic quart container with some meat but without the pasta. You can reheat frozen soup in the microwave or thaw it first in the refrigerator.
Removing some of the carrots after about 35 minutes of cooking ensures they won't be mushy. Reserve them to serve in the bowls of soup later or make Italian Carrot Salad with them.
How to use leftover soup meat: My family often stirs olive oil, red wine vinegar, salt and pepper into the extra shredded beef to make a meat salad. It's delicious with Italian bread.
Can you make this in a slow cooker? You can, but it will be greasy the first day, since it's not easy to remove the foam (meat protein) the way you can when the soup boils on the stove. However, after overnight refrigeration, the grease will solidify at the top and you can remove it. The soup will taste much better the second day.
Note that to make this recipe fit in the Crock-Pot, you will either need to reduce the liquid to six cups or cut up the meat and veggies before adding them. (You can add them all at once.) Make sure the Crock-Pot has at least an inch of space at the top so it doesn't overflow. You don't need to strain the soup or remove the carrots early. Cook the pasta separately.
Can you make this in the Instant Pot? I haven't tried it but suspect we might have to adjust the recipe to fit into a 6-quart pressure cooker. Also, the meat would probably have to be cut into chunks first. I'll update the recipe when I try it.