Step #1 Heat 2 tbsps oil in a 5- to 6-quart pot over high heat until hot but not smoking, then cook onions, stirring every once in awhile, garlic, until onions are pale golden, celery, & carrots, about 4 mins.
Step #2 Stir in broth & water & bring to a boil, covered.
Step #3 Meanwhile, until browned all over, turning every once in awhile, heat remaining 2 tbsps oil in a 12-inch heavy skillet over high heat until hot but not smoking, then sauté meatballs (do not thaw if frozen), about 3 mins.
Step #4 Add meatballs to soup along with beans & briskly simmer, covered, until vegetables are tender & meatballs are heated through, stirring every once in awhile, about 15 mins.
Step #5 Stir in spinach, salt, cheese, uncovered, & pepper & let simmer, until spinach is wilted, about 1 min.