Turn on a gas grill to medium-high or light a charcoal grill and let the coals burn until quite hot and covered with white ash.
Generously sprinkle the meat on all sides with salt and pepper. Lay the short ribs on the grill and sear on all sides, until richly browned, 4 or 5 minutes total depending on the heat of your fire. Remove to a rimmed baking sheet.
Break the stems off the chiles, then roll them between your fingers to loosen the seeds. Break the chiles in half and shake out all the seeds that come free easily. Set a large (7- to 8-quart) Dutch oven over medium-high heat. When the pot is hot, pour in the oil. Add the chiles and stir for 10 to 15 seconds, until they are noticeably darker and aromatic. Remove to a small plate, draining as much oil as possible back into the pan.
Add the onion to the pan and cook, stirring regularly, until golden, about 7 minutes, then stir in the mushrooms and cook another couple of minutes. Add the beer, broth, garlic, thyme, tomatoes, beans, toasted chiles, 1 1/2 teaspoons salt and a generous 1/2 teaspoon black pepper. Return the short ribs (and any juices that have collected around them) to the pan, nestling them into the liquid.
Either turn the grill burner(s) in the center to medium-low or bank the coals to the sides for indirect cooking. Set the pot in the center of the grill and cook until the shortribs are fork-tender, 2 to 2 1/2 hours. As necessary add additional charcoal to keep the fire medium hot.
Carefully remove the short ribs to a deep serving platter. Discard the garlic and thyme sprigs from the braising liquid. Using a slotted spoon, spoon the beans and mushrooms around the short ribs. If a lot of fat has rendered from the short ribs, ladle it from the top of the sauce. Taste the sauce and season with additional salt if you think necessary, then ladle it over the ribs and beans. You’re ready to serve.
Oven Alternative: This dish can be made very successfully in the oven. In the Dutch oven in a little oil, brown the meat, then braise everything together, covered, for about 2 1/2 hours at 325 degrees.
Working Ahead: The whole dish can be made a day or so ahead, cooled, covered and refrigerated. When making the dish ahead, I like to remove the meat to a separate container to ensure that its juices don’t overwhelm the other flavors of the dish. Reheat, covered with foil, in a 350 degree oven for 30 to 45 minutes.