Soak beans 8 hours or overnight in water to cover by 1 inch. Drain.

Heat 1/4 cup olive oil in a large pot over moderate heat. Add onion, carrots, garlic, parsley and bay leaves. Saute until vegetables are slightly softened, about 10 minutes.

Add drained beans, the chicken broth and 2 cups water. Season with salt and pepper. Bring to a simmer, cover and adjust heat to maintain a bare simmer. Cook, stirring occasionally, until beans are tender, 45 minutes to 1-1/2 hours, depending on age of beans. Remove the bay leaves. In a food processor or blender, puree 4 cups of the beans and vegetables with some of their liquid. Return the puree to the pot and stir well. If needed, thin soup with water.

Wash greens well, removing any bruised leaves, thick ribs or stems that are thick or tough. You should have about 1/2 lb trimmed greens. Stack leaves a few at a time, roll into a log and slice crosswise into 1/4-inch-wide ribbons. Bring a large pot of salted water to a boil over high heat. Add greens and boil until tender--a couple of minutes for young dandelion and turnip greens, longer for kale or collard greens. Drain, reserving about 2 cups of the cooking liquid. Stir greens into soup. Cover and simmer an additional 5 minutes. Thin soup, if needed, with reserved cooking liquid. Taste and adjust seasoning.

Serve in warm bowls, topping each portion with a drizzle of olive oil and a sprinkle of cheese.