Ingredients 3 cups shelled fresh shelling beans (from about 2 pounds in pods), or 2 cans (each 14.5 ounces) cannellini, borlotti, or butter beans, drained and rinsed 1/2 bulb fennel, plus 1 cup chopped (from 1 large bulb) 1 whole peeled shallot, plus 1 cup chopped (from 1 large) 1 halved carrot, plus 2 to 3 more, peeled and sliced into 1/4-inch rounds (1 cup) 3 cloves garlic, lightly smashed, plus 1 tablespoon minced Kosher salt (we use Diamond Crystal) and freshly ground pepper 2 pounds large tomatoes, such as beefsteak (3 to 4) 2 tablespoons extra-virgin olive oil, plus more for drizzling 1 tablespoon tomato paste 8 ounces pole beans, such as green beans or Romanos, trimmed and cut into 1-inch pieces (2 cups) 4 cups water or vegetable stock (preferably homemade) 3 ounces short, flat pasta, such as stracci Toscani or lasagna noodles broken into 1-inch pieces 1 to 2 teaspoons red-or white-wine vinegar 4 cups dark leafy greens, such as dandelion, escarole, or lacinato kale, roughly chopped Chopped fresh parsley and red-pepper flakes, such as Calabrian, for serving Preparation Combine shelled beans, fennel bulb, whole shallot, halved carrot, and smashed garlic cloves in a medium pot; add water to cover by 2 inches....