Preheat the oven to 300°F (150°C).
Season the meat with salt and pepper. In a large Dutch oven, warm the olive oil over medium-high heat. Add the meat and sear, turning once, until browned, about 5 minutes per side. Transfer the meat to a plate.
Add the celery, onion, carrot, and garlic to the oil and beef drippings in the Dutch oven and sauté until the vegetables are softened and lightly browned, about 10 minutes. Add the tomato paste and cook, stirring often, for about 5 minutes.
Add the wine, stir to scrape up the brown bits on the bottom of the pan, and cook until the wine is nearly evaporated, about 10 minutes.
Add the meat, thyme, parsley, rosemary, bay leaves, peppercorns, demi-glace, and enough water to almost cover the meat. Cover, place in the oven, and braise for 1 1/2 hours.
Uncover the pot, rotate it 180°, and continue to braise until the meat is fork-tender and nearly falls off the bone, about 1 hour more.
Using tongs, transfer the meat from the pot to a cutting board and set aside until cool enough to handle. Return the pot to the stovetop and simmer over medium heat until the liquid is reduced by half.
Meanwhile, pull all of the meat from the bones, if not using stew meat, and shred it. Discard the bones and any gristle. Strain the braising liquid through a fine-mesh sieve and discard the solids.
Warm a large skillet over high heat and add the olive oil. When the oil is hot, add the carrots and sauté until slightly charred, about 3 minutes. Add the reserved meat, chopped fresh herbs, and wine and cook until the wine has nearly evaporated. Add the reduced braising liquid and simmer until the mixture has a nice sauce-like consistency.
Season with salt and pepper to taste. Add the pasta to the sauce. Cook, gently stirring, until the pasta is al dente and well coated with the sauce, about 3 minutes. If the sauce is too thick, add a little pasta cooking water to thin it.
Divvy the red wine pasta, meat, and vegetables among warmed plates. Top each portion with the Parmigiano-Reggiano or pass it at the table.