Preheat oven to 325°F. Pat the short ribs completely dry with paper towels and season aggressively on all sides with kosher salt and pepper. Dry meat sears better — moisture is the enemy of a good crust.
Heat olive oil in a large Dutch oven over high heat until smoking. Sear the short ribs in batches for 3-4 minutes per side until a dark, hard crust forms on all sides. Do not rush this and do not crowd the pot — you are building the flavor foundation for the entire dish. Remove the ribs to a plate.
Reduce heat to medium. Add the diced onion, carrots, and celery to the pot with all the rendered fat and fond. Cook for 6-8 minutes, stirring occasionally, until the vegetables are softened and starting to caramelize. Add the smashed garlic and cook for 60 seconds.
Add the tomato paste and stir it into the vegetables for 2 minutes — cooking the paste removes the raw tinny taste and deepens the color of the sauce. You should see it darken on the bottom of the pot.
Pour in the red wine and stir, scraping every bit of fond from the bottom of the Dutch oven. This is where a huge amount of flavor lives. Let the wine reduce by half, about 3-4 minutes.
Add the crushed San Marzano tomatoes, beef broth, bay leaves, thyme sprigs, and rosemary sprig. Stir to combine and bring to a simmer.
Nestle the seared short ribs bone-side up back into the liquid. The braising liquid should come about three-quarters up the sides of the meat — not fully submerged. Cover with a tight-fitting lid and transfer to the 325°F oven.
Braise for 3 to 3.5 hours, checking once at the 2-hour mark. The short ribs are done when the meat is falling off the bone and shreds with zero resistance when you press it with a fork. The house will smell unbelievable.
Remove the short ribs from the braising liquid and set on a cutting board. Discard the bones, any large fat chunks, bay leaves, and herb stems. Shred the meat into large, rustic pieces with two forks — do not shred too finely. You want chunks, not threads.
Strain the braising liquid through a fine mesh strainer, pressing on the vegetables to extract every bit of flavor. Discard the solids. Return the strained liquid to the Dutch oven and bring to a simmer over medium heat. Let it reduce for 10-15 minutes until it coats the back of a spoon like a light gravy. Skim any fat from the surface.
While the sauce reduces, cook the pappardelle in a large pot of aggressively salted water until 1 minute short of al dente. Reserve 1 cup of starchy pasta water before draining.
Add the shredded short rib meat back into the reduced sauce. Toss in the drained pappardelle and stir gently over medium-low heat for 60-90 seconds, adding splashes of pasta water to loosen until the sauce clings to every ribbon. Remove from heat and stir in the cold butter and half the Parmigiano-Reggiano until melted and glossy.
Serve in warm bowls. Top with the remaining grated Parmigiano-Reggiano, chopped fresh parsley, a crack of black pepper, and flaky sea salt. Put crusty bread on the table for mopping up the sauce — anything left in the bowl is a crime.