In a tall 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 spoon to skim off the foam that comes to the surface and discard it.
While the beef is cooking, prepare your vegetables. Peel and quarter your 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 smash your garlic. Rinse your basil and remove the leaves.
After 20 minutes, add your 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 to shred or cut into chunks. Strain the soup into another pot (place a strainer over the empty pot and pour your soup through the strainer to remove the vegetables.) Discard the peppercorns. The vegetables in the strainer will be mushy, but you can save them to eat with salt and pepper if you wish.
Taste the soup to see if it needs additional salt and pepper. Place the beef back into the strained soup.
For the pasta, place half of the soup broth in a pot and bring it to a boil over high heat. Slice up your reserved carrots and add them to the boiling broth with the pasta. Cook the pasta until it is al dente (check it about 3 minutes before the al dente time on the package.) Turn off the heat, but do not drain the pot.
To serve, scoop some pasta and carrots into each bowl and ladle the broth and meat over it. Top each serving with freshly grated Parmesan and have salt and red pepper flakes at the table for extra seasoning.
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 leftovers should be good for four days.