Preheat the oven to 325°F. Position a rack in the lower third of the oven.
Pat the short ribs dry with paper towels and season generously on all sides with salt and pepper. Set aside while you prepare the vegetables.
Heat a Dutch oven over medium-high heat and add a small amount of olive oil. You only need enough to lightly coat the bottom, as the short ribs will render plenty of fat.
Working in batches to avoid crowding, add the short ribs and sear for 3-4 minutes per side, until deeply browned but not burned. Transfer to a plate and set aside.
Reduce the heat to medium. Add the shallots, carrots, celery, and garlic to the pot. Season with salt and pepper and cook, stirring occasionally, for 7-8 minutes, until softened and lightly caramelized.
Stir in the tomato paste and cook for 1-2 minutes, coating the vegetables and letting the paste deepen in color. Sprinkle in the flour and cook for another 1-2 minutes, stirring constantly.
Pour in the wine, broth, balsamic vinegar, thyme, and bay leaves. Season lightly with salt and pepper, keeping in mind the sauce will reduce significantly. Increase the heat to medium-high and bring the mixture to a simmer, scraping up any browned bits from the bottom of the pot.
Return the short ribs to the pot, nestling them into the liquid so they are mostly submerged. Bring back to a gentle simmer, then cover and transfer to the oven.
Braise for 4-5 hours, until the meat is very tender and the liquid has reduced by about half.
When the short ribs are nearly finished, bring a large pot of salted water (about 6 quarts water with 2 tablespoons salt) to a boil. Cook the noodles according to package directions, then drain well.
Gently spoon the sauce over the short ribs to keep them moist, then transfer them from the pot to a plate and set aside. Stir the cooked noodles into the sauce and vegetables, tossing gently to coat. Return the short ribs to the pot or serve them on top of the noodles.
Serve in bowls, finished with chives and parmesan if desired. If you prefer, remove the bones, shred the meat, and stir it back into the sauce before serving.