Stirring in just a little bit of miso paste will take your minestrone from mild to mega-flavorful.
In a medium saucepan, heat the oil over medium-high. Add the leeks and carrots. Cook, stirring occasionally, until the leeks soften, about 5 minutes. Stir in the pesto and cook, stirring constantly, for 20 seconds. Add the zucchini and 4 cups of water. Season. Bring to a boil, then reduce to a simmer and cook until the vegetables are tender, about 4 minutes.
Meanwhile, bring a medium saucepan of water to a boil. Salt the water, add the pasta, and cook until al dente. Drain the pasta.
Stir the pasta, spinach, beans, and peas into the soup. Simmer until the spinach wilts, about 2 minutes. Remove the soup from the heat.
In a small bowl, whisk the miso with 1/4 cup of warm water; stir into the soup. Season the soup with pepper. Ladle into bowls and top with a generous sprinkle of Parmesan.