Sort lima beans, discarding debris. Rinse beans, put in a 3- to 4-quart pan, and add 1 1/2 quarts water. Bring to a boil over high heat; remove from heat and let stand at least 1 hour or overnight. Drain beans, add 1 1/2 quarts water, and bring to a boil over high heat. Cover, reduce heat, and simmer, stirring occasionally, until beans are tender to bite, about 20 minutes. Drain beans; reserve liquid.
Meanwhile, trim and discard stem end and tough green top from leek; peel off and discard tough outer layer. Cut leek in half lengthwise and hold under cold running water, flipping layers to rinse well. Thinly slice crosswise.
In an 8- to 10-quart pan over medium heat, frequently stir pancetta in 2 tablespoons olive oil just until limp, about 2 minutes. Add leek, onion, celery, and parsley; stir often over medium-high heat until vegetables begin to brown, 10 to 15 minutes.
To pan, add 2 1/2 cups reserved bean-cooking liquid (save remainder for other uses or discard), drained beans, broth, tomatoes with juice, potatoes, and carrots. Bring to a boil over high heat; cover, reduce heat, and simmer, stirring occasionally, about 10 minutes.
Add cabbage, Swiss chard, green beans, and zucchini; cover and simmer, stirring occasionally, until all vegetables are very tender to bite, 20 to 25 minutes longer.
Ladle soup equally into bowls; spoon 1 tablespoon olive oil and about 1/2 tablespoon pesto alla genovese into each portion. Serve with remaining pesto and salt and pepper to add to taste.