Heat oil in a large pot over high heat. Saute onion, carrots, celery, and garlic until onion is pale golden, about 4 minutes. Pour broth and water into pot; cover the pot and bring mixture to a boil.
Stir beans and meatballs into broth mixture; bring to a boil, reduce heat to medium-low, cover the pot, and simmer, stirring occasionally, until vegetables are tender and meatballs are heated through, about 20 minutes.
Mix spinach, Asiago cheese, salt, and pepper into soup; simmer until spinach wilts, about 1 minute.