Peel and finely chop the onion. Peel and finely dice enough carrot to measure about ¼cup. Peel the back of the celery to remove the tough strings and finely chop enough to measure about ¼cup.
Put the olive oil, butter and onion in a saucepan together with the onion, carrot, and celery. Sauté over medium heat until the vegetables begin to color. While the vegetables are Sautéing, dice the prosciutto. When the vegetables are ready, add the prosciutto and cook until it loses its raw color, about 2 minutes. Add the tomatoes, season with salt and pepper, and cook until they are no longer watery and have separated from the oil, 20 to 25 minutes.
Fill a pot for the pasta with about 6quarts of water, cover, and place over high heat.
Add the beans and ½cup of water to the saucepan and cook for about 5 minutes. Scoop up about half of the beans and purée them through a food mill or in a food processor then return them to the pot. Add the parsley and cook for 2-3 more minutes. The sauce should be moist enough to pour out of a spoon while leaving a thick coating on it. If necessary add a little more water or cook a little longer if it is too watery.
Add 2tablespoons salt to the boiling pasta water, put in the cavatappi, and stir well. Cook until al dente then toss with the sauce and the freshly grated Parmigiano-Reggiano cheese. Grind a little fresh pepper and drizzle a little olive oil over each serving.