Position a rack in the lower third of the oven and heat the oven to 300°F.
Soak the mushrooms in 2 cups of warm water for 20 to 30 minutes. With a slotted spoon, transfer the mushrooms to a cutting board and chop them finely. Line a strainer with a coffee filter or two layers of paper towels and strain the mushroom-soaking water into a bowl to get rid of any grit. Set aside both mushrooms and liquid.
Put one can of tomatoes and their juices in a food processor and process until puréed. Using a spatula or the back of a ladle, press the purée through a medium-mesh sieve set over a bowl to remove the seeds. Purée and strain the other can of tomatoes.
Heat 1/4 cup of the oil in a 7- to 8-qt. Dutch oven over medium-high heat. Season the ribs and beef chuck with kosher salt and pepper and add them to the hot oil. Cook, turning as necessary, until the meat is golden brown on all sides, about 10 minutes. Transfer to a large plate, discard the fat, and clean the pan with paper towels.
Heat the remaining 1/4 cup oil in the pan over medium heat. Add the onion, carrot, celery, garlic, and pancetta and cook, stirring frequently, until the vegetables are lightly golden and soft, 7 to 8 minutes. Add the mushrooms and parsley and stir for about 1 minute to blend the ingredients.
Return the meat to the pan and stir to coat with the savory base. Raise the heat to high and add the wine. Cook, stirring occasionally, until the wine is reduced approximately by half, about 5 minutes. Add the tomatoes and 1/2 cup of the reserved mushroom-soaking water. Season with 1/2 tsp. salt and 1/4 tsp. pepper. Stir until the liquid begins to simmer.
Turn off the heat, cover the pan tightly with a lid or heavy-duty aluminum foil, and put it in the oven. Cook, turning the meat every half hour, until the meat is fork tender and the ribs begin to fall off the bone, about 2-1/2 hours.
Remove the pan from the oven and transfer the meat (including any that has fallen off the bone) to a cutting board. Use a ladle to skim the fat off the surface of the sauce. When the meat is cool enough to handle, pull the meat off the ribs. Discard the bones and any fat and connective tissue. Finely chop all the meat. Stir the meat back into the sauce and simmer on the stovetop over medium heat, stirring occasionally, to allow the flavors to meld and the sauce to thicken slightly, about 10 minutes. Adjust the seasoning with salt and pepper to taste.
The last step--toss it with pasta:
Once you've made your rich ragù, all that's left is to combine it with pasta. To serve four to six people, you'll need 4 cups ragu, 1 Tbs. unsalted butter, 1 lb. dried or fresh pasta (cooked and drained), and 1/2 cup freshly grated Parmigiano-Reggiano or Pecorino Romano.
Heat the ragù (either in the Dutch oven you used to cook it or in a 12-inch skillet, if reheating) over medium-high heat. Add the butter and then pour in the pasta and Parmigiano or pecorino. Toss over medium-high heat until the pasta and sauce are well combined. Serve immediately.
Short, full-bodied dried pastas like rigatoni and orecchiette work great with ragù, because their nooks and ridges capture the sauce. If you want to use fresh pasta, a wide shape like papperdelle can stand up to a hearty sauce. And a ragù is a good excuse to cook gnocci, too.
The earthiness, intense fruit, and bright acidity of an Italian Barbera would be a good match for this ragù. Try the 2004 Michele Chiarlo Barbera d’Asti ($14) or the 2004 Moccagatta Barbera d’Alba ($19).