Heat oil in a large, deep saute pan or soup kettle over medium-high heat. When pan is hot, add sausages; cook, turning once or twice, until well-browned on all sides, about 5 minutes. (Sausage will not be fully cooked at this point.) Remove from pan. When cool enough to handle, cut into slices 1/4-inch thick.

Add prosciutto, onions, carrots, celery and thyme to the empty skillet; cook, stirring often, until well browned, 8 to 10 minutes. In a small bowl, mash one can of beans with a fork into a chunky puree. Add broth, whole and mashed beans, and sausage; cover and bring to a simmer. Reduce heat to medium-low; simmer, partially covered, to blend flavors, 20 minutes. Let rest 10 minutes; serve.