Trim the excess fat off of the short ribs with a knife and set aside.
Heat the oil in your stock pot and brown the short ribs on both sides. Remove them from the pan onto a plate. Add in your vegetables and sauté for a few minutes until they are slightly cooked and the onions are translucent and your tomatoes have popped. Add the short ribs back in with the juices from the plate.
Cover the meat and vegetables with your water and bring to a boil. Skim any foam that floats to the surface with a spoon and discard. Once boiling add bay leaves. Turn the flame down to medium and cook stirring occasionally until the meat is tender and begins to separate from the bone, about 1 hour and 30 minutes to 2-4 hours.
When the soup has about 15 minutes to go, bring pasta water to a boil. Break the pasta by hand into 2 inch pieces and drop them into salted boiling water. Or cook your small pasta according to the package directions. Cook until al dente.
Drain the pasta and add it to a bowl. Ladle the broth on top, leaving the meat for seconds. You can add in the vegetables or you can just serve the pasta with the broth. I like to eat some of the potatoes with the meat after we eat the pasta. I also used to smash up the carrots and potatoes and give them to my daughters with her pasta when they were little. They loved it!
Serve with grated parmigiano reggiano cheese and crusty bread. Enjoy the meat after the pasta is served with the broth and finish the meal with some homemade fries! ENJOY!