In a medium soup pot, heat the EVOO, 2 turns of the pan, over medium heat. Stir in the carrot, celery, onion, garlic and oregano; season with salt and pepper. Cover the pot and cook, stirring occasionally, until slightly softened, about 5 minutes. Add the diced and stewed tomatoes and the chicken stock and bring to a simmer; cook until the vegetables are tender, about 20 minutes.
Meanwhile, bring a large saucepan of salted water to a boil. Add the Broccolini and cook until crisp-tender, 3 to 4 minutes. Drain.
Using an immersion blender, puree the soup. Stir in the Broccolini and basil; season with salt and pepper. Serve with lots of cheese and bread.