Trim away any excess fat from the top of each rib down to the first layer of meat, but don’t remove the silverskin or tough-looking tissue that hold the ribs together or onto the bone. Pat the ribs dry with paper towels, and season them with salt and pepper. Heat the oil in a 5- to 6-quart Dutch oven (or other heavy pot with a lid) over medium heat until hot. Add only as many ribs as will fit without touching, and brown them on all sides, turning with tongs until nicely browned, 3 to 4 min. per side. Transfer to a platter, and continue until all the ribs are browned.

Pour off and discard most of the fat from the pot. Return the pot to medium heat, and add the pancetta or bacon. Cook, stirring occasionally, until the pancetta or bacon has rendered some of its fat but is not crisp, about 3 minutes. Add the onion, celery, carrot, mushrooms, and garlic, and season with salt and pepper. Cook, stirring occasionally, until softened, about 8 min. Add the tomato paste and red pepper flakes, and cook for 2 min., stirring. Add the wine, increase the heat to medium-high, stir to scrape up any browned bits on the bottom of the pot, and bring to a boil. Add the tomatoes and their juices and bring to a simmer.

Return the short ribs (and any juices) to the pot in a single layer, nestling them into the sauce. Crumple a large sheet of parchment and smooth it out again. Arrange it over the pot, pressing it down so it nearly touches the ribs, allowing any overhang to extend up and over the edges of the pot. Set the lid in place, and transfer to the oven. Braise, turning the ribs with tongs every 45 min., until fork-tender and the meat is pulling away from the bone, about 2-1/2 hours.

Use tongs or a slotted spoon to carefully transfer the ribs to a shallow baking dish or wide bowl. Set aside until just cool enough to handle. Skim any visible surface fat from the braising liquid and bring to a gentle simmer over medium heat. If the tomatoes haven’t broken up, break them up with a wooden spoon. If the sauce seems watery, increase the heat to medium-high and simmer to thicken. If not, keep it warm. Taste for salt and pepper.

When the beef is cool enough to handle, remove the meat from the bones and the bands of connective tissue that encircled the bones. Chop the meat into bite-size chunks or pull into large strips. Return the meat to the sauce, and keep warm until ready to serve, or if making ahead, cool and refrigerate.