Trim the celery stalks (see Notes) and cut them into short pieces. Boil the pieces in well salted water until perfectly tender, about 10-15 minutes. If using the leaves of the celery, add them about halfway through the cooking.
Drain the celery well and let it cool in a colander. When cool, squeeze as much of their liquid out of them as you can out and mince well.
Add the minced celery to a large mixing bowl and add the eggs, grated cheese, salt and a good grind of pepper. Mix everything well, then mix in the breadcrumbs, as much as you need to bind everything together into a kind of 'dough'. Let this rest for about 15-20 minutes.
Form the celery mixture into balls about the size of walnuts. A small ice cream scoop makes short work of this task. When you're done, roll the celery balls in more breadcrumbs so they are well coated.
Shallow fry the celery balls into moderate hot oil until they are golden brown all over, making sure they're well spaced. Proceed in batches if need be. As you go, set the fried celery balls on paper towels or a baking rack to drain of their excess oil.
Serve your polpette di sedano while they're still warm, if you like on a bed of tomato sauce, or at room temperature if you're enjoying them as finger food.