Pour chicken broth into a large pot and bring to a boil. Add carrots, celery, garlic and salt. Reduce the heat to medium-low and simmer for 30 minutes. Meanwhile, in a mixing bowl, combine breadcrumbs, cheese and nutmeg. Make a well in the center and add eggs; mix until thoroughly combined. The dough should be damp but firm. If necessary, incorporate additional breadcrumbs or a few drops of water; if the dough is too wet or too dry it will crumble. To form the passatelli, pinch off a piece of dough and roll it about 1/4 inch thick, then break it into 1/2 inch lengths. Continue forming the rest of the dough in the same manner. Bring the broth back to a boil, add the passatelli and parsley and cook for 5 minutes. Ladle the soup into bowls, sprinkle with a little Parmesan and serve.