Heat a medium soup pot over medium high heat. Add extra-virgin olive oil, 2 turns of the pan. Add onion, potato and zucchini and season with salt and pepper. Cook 5 or 6 minutes alone, stirring frequently, then add the canned beans and the chicken stock. Cover the pot and bring the soup up to a boil.
Preheat a panini grill or a heavy bottomed pan over medium heat. Build 4 panini: spread roll or bread with a couple ounces of cheese, then top with a few leaves of arugula and 2 slices of prosciutto for each sandwich. Place panini in sandwich press or onto hot, heavy bottomed pan. Top with another heavy skillet or a brick wrapped in foil to press sandwiches. Press sandwiches until crisp on both sides, 2 to 3 minutes on each side.
When the soup is at a boil, remove the lid. Stir in the noodles and green beans and reduce heat to low. Simmer soup an additional 5 to 6 minutes then remove it from the heat.
Combine basil, garlic, nuts and cheese in the food processor and turn it on. Stream in the extra-virgin olive oil and process until pesto is nearly smooth.
Scrape the pesto into a small bowl. Add a ladle of the soup broth to thin the pesto, then add the pesto to the soup. Stir to combine and serve soup immediately with panini, for dipping.