To make the pappardelle in your slow cooker, see the Slow Cooker Variation below. To make the pappardelle on your stovetop, season the meat with salt and pepper. (It can be cut into large 2-inch chunks if it's beef, venison, or wild boar; left whole if it's squab; or cut into 5 or 6 pieces if it's rabbit.) Place a Dutch oven over medium-high heat. Add a little olive oil and the meat, being careful not to crowd the pot, and cook until golden brown on all sides. It may be necessary to work in batches; if so, return all the meat to the pot after it's browned.
Add your herbs, onions, garlic, carrot, and celery to the meat in the pot, reduce the heat, and continue to cook, stirring occasionally, until the vegetables have softened, about 5 minutes.
Pour in the wine and simmer just until the liquid has almost completely cooked away but has left you with a fantastic color and fragrance. Add the plum tomatoes, the pearl barley, and just enough water to cover the meat by about 1/2 inch.
Gently place the parchment circle directly on the surface of the liquid. Put a lid on the pot as well to retain as much moisture as possible while cooking.
Reduce the heat to low and gentle simmer for 2 to 3 hours. The exact timing will depend on the tenderness and type of meat. You'll know that the meat is ready when you can literally tug on it with a fork and it easily pulls apart in strands.
Season the beef ragu with salt and pepper to taste and let it cool slightly before removing the meat from the pan. Using 2 forks, pull apart all the chunks of meat, throwing away any gristle or bits of undissolved fat. Skim any fat from the surface of the braising liquid. Return the shredded meat to the pot and place over low heat.
Bring a pot of heavily salted water to a boil and cook your pappardelle for 3 minutes if using fresh pasta or according to the package instructions if using dried. Once it's cooked, drain it in a colander, saving some of the cooking water.
Remove the meat sauce from the heat and stir in the butter and Parmesan along with a little of the reserved pasta cooking water—this will make it juicy and shiny.
Toss the beef ragu with your drained pasta and, if desired, serve sprinkled with a little finely chopped fresh rosemary and some more grated Parmesan. Serve immediately.