Place the green beans, carrot, and potatoes in a large saucepan. Add 8 cups water and 2 teaspoons sea salt, bring to a boil and cook, covered, until the carrots are nearly tender through, about 12 minutes. Add the zucchini and beans, and let cook for another 3 to 4 minutes. Add the angel hair pasta and cook until it is al dente (almost done), 3 to 4 minutes. Remove the soup from the heat. Mix in half the pesto, and stir well to combine. Serve soup with a small dollop of pesto in each bowl.