In a 4- to 5-quart pan over medium-high heat, combine oil, onion, carrot, celery, and garlic; stir often until onion is limp, about 5 minutes.
Add broth, cover, and bring to a boil over high heat.
Stir penne pasta into broth and cook 7 minutes.
Meanwhile, trim stem ends and pull any strings from beans. Rinse beans and cut into 2-inch lengths. Add beans to pasta; cook until both are tender to bite, 5 to 6 minutes longer. Add basil and mix.
Ladle into bowls and scatter prosciutto evenly over portions. Add parmesan cheese and salt and pepper to taste.