How to Make It
In a very large stockpot, combine the hen, veal shank, pork ribs, celery, carrots, garlic and dried chiles. Add enough of the water to just cover the meats and bring to a simmer over moderately low heat. Add a large pinch of salt and pepper and reduce the heat to low. Cover partially and cook until the broth is very flavorful, about 3 hours.
Meanwhile, in a large pot of boiling salted water, cook the fresh greens in 2 or 3 batches until just tender, 5 to 8 minutes; if you are using a combination of greens, cook each variety separately. Drain the greens and when they're cool enough to handle, squeeze them dry. Coarsely chop the greens.
When the broth is done, remove the hen, veal shank and pork ribs. Strain the broth through a fine sieve into a large bowl. Remove all the meat from the hen, veal shank and pork ribs and cut into bit-size pieces.
Skim the fat from the surface of the broth and pour it into a clean stockpot, stopping when you reach the solids at the bottom. Add the meats and greens to the broth and bring just to a simmer. Season with salt and pepper and serve piping hot in rimmed soup plates. Pass the cheese at the table.