To prepare battuto, combine first 6 ingredients in a food processor; pulse until mixture is almost a paste.
To prepare soup, heat oil in a Dutch oven over medium-high heat. Add battuto; sauté 3 minutes. Add broth and next 4 ingredients (broth through rosemary); bring to a boil. Cover, reduce heat, and simmer 5 minutes. Add green beans and next 5 ingredients (green beans through spaghetti). Simmer, uncovered, 20 minutes or until vegetables are tender. Discard rosemary. Top with cheese.