Put a 4 quart saucepan over a medium-high flame and heat the oil. Add the carrots, shallot, celery rosemary and bay leaves to the pan. Cool until the vegetables have softened, about 5 minutes.
Add the prosciutto to the pan and cook until the prosciutto begins to slightly crisp, about 3-4 minutes.
Take about 1 cup of the chickpeas and mash them with a spoon. Add them to the pan along with the whole chickpeas. Cool for 5 minutes and add the water and parsley. Bring to a boil and cook uncovered over medium heat until the mixture begins to thicken, about 20 minutes. Taste for seasoning and season with salt and pepper as desired. Add the parsley.
Bring a large pot of generously salted water to a boil and cook the pasta for 5 minutes less than the package instructions. The pasta should be hard.
Drain the pasta and add it to the pan with the chickpeas. Cook until the pasta is al dente, about 2-3 minutes.
Serve in warm bowls with an extra drizzle of extra virgin olive oil and a sprinkle of grated Parmigiano Reggiano cheese.