For the Fish Minestrone: Warm the olive oil in a large pot over medium-low heat. Add the leeks, carrots, zucchini, and green beans. Stir to combine and cook until tender, about 10 minutes. Add the beans and the chicken broth. Cook until the broth is simmering and the beans are warmed through, about 5 minutes. Season the fish filets with the salt and pepper and add them to the simmering soup. Simmer until the fish is cooked, about 7 minutes depending on the thickness of the fish. Season the soup with salt and pepper.
For the Herb Sauce: While the fish cooks, combine the parsley, oregano, garlic, and red wine vinegar in a food processor. Pulse the machine until the herbs are almost a paste. Add the olive oil in a steady stream while the machine is running. Season with salt and pepper.
To serve, ladle the soup into bowls and top each serving with a spoonful of the herb sauce. Serve immediately.