Beat the egg(s) in a mixing bowl, then add the remaining ingredients (except for the broth). Mix with a spoon until the egg is incorporated, then work the mixture with your hands into a homogenous ball of dough.
Wrap the dough in film wrap and let rest for at least an hour at room temperature.
When you're ready to cook, bring broth to a gentle simmer while you make your passatelli.
Take one portion of the passatelli dough and put it in a potato ricer with large holes, then squeeze it through the holes onto a cutting board, cutting them into short lengths. (NB: Alternatively, you can squeeze the passatelli directly into the simmering broth, cutting them as needed as you go.)
Transfer the passatelli gingerly into the broth and simmer for about 1-2 minutes. The passatelli will be done when they have expanded slightly and turned a lighter shade of yellow. Do not stir.
Serve immediately, with additional grated cheese on the side for those who want it.
