Bring the broth to a fairly brisk simmer (but not a rolling boil).
While the broth is coming up to heat, in a mixing bowl scramble the egg together with the grated cheese, salt and pepper and, if you like, one or more of the optional flavorings. Whisk well until perfectly smooth and homogenous.
When the broth is at the simmer, pour the egg mixture into the broth in one go, all the time whisking (or stirring vigorously with a wooden spoon) in a single direction. After a minute or two, the egg mixture will separate into little curds.
Allow the soup to simmer for a couple of minutes more and serve hot, with additional grated cheese on the side for those who care for it.