Pre-heat a large dutch oven, or large pot with a lid, to medium-high heat. Season the short ribs with salt and pepper, add a splash of oil to the pan, then sear the ribs on all sides until golden brown. You may need to do this in batches. Remove the short ribs from the pan and set aside.
Turn the heat down to medium and sauté the carrots, onions and garlic until soft and fragrant. Next, stir in the tomato paste, cook for 2-3 minutes, then deglaze with the wine. Let the wine reduce by half.
Add the short ribs back into the pan along with the chicken stock and thyme. Bring to a simmer then cover and reduce the heat to low. Let the ribs gently simmer on the stovetop, or in a preheated 350F oven for 2.5-3 hours. Check them every hour or so, stirring the bottom of the pot to ensure the braising liquid isn't scorching. Use a ladle to skim away the excess fat that forms at the surface.
The ribs are done cooking when the bones are falling out and the meat is fork-tender. At this point remove them from the pot and set aside until they are cool enough to handle. Discard the thyme stems from the pot as well, then use an immersion blender to blend the sauce and break up the carrots and onion. If you don't have an immersion blender you can just blend 1 cup of the braising liquid in a regular blender. Skipping blending all together is fine too, the sauce will just be a little bit thinner.
Remove and discard the bones and any excess fat or cartilage from the ribs then shred the meat with your hands or two forks and add it back into the braising liquid.
Turn off the heat then fold in the mascarpone cheese. Lastly, add the cooked pappardelle to the pan and toss it with the ragù. Divide between bowls and serve immediately with lots of shaved parmesan on top.