Heat olive oil in a large, deep sauté pan or soup kettle over medium-high heat. Add sausages and cook, turning once or twice, until well-browned on all sides, about 5 minutes. The sausages will not be fully cooked at this point. Remove from the pan and let them cool. Once cool enough to handle, cut them into slices about 1/4-inch thick.

In the same pan, add minced prosciutto, onions, carrots, celery, and dried thyme. Cook, stirring often, until the vegetables are well browned, about 8 to 10 minutes.

In a small bowl, mash one can of beans with a fork into a chunky puree.

Add the mashed beans, whole beans, chicken broth, and sliced sausage to the skillet. Cover and bring to a simmer over medium heat.

Reduce the heat to medium-low and simmer, partially covered, to blend the flavors for about 20 minutes.

Let the soup rest for 10 minutes before serving.