Place 2 tablespoons of the olive oil, the onion, and garlic in a large heavy-bottomed soup pot over medium heat. Cook slowly, stirring occasionally, until the onion is translucent, about 3 minutes. Add the zucchini and the celery and keep cooking, stirring occasionally, for 2 minutes more. Add another tablespoon of the olive oil and the cauliflower and cook 5 minutes, stirring occasionally, as the vegetables slowly begin to soften. Stir in the green beans, cabbage, and remaining olive oil and cook, stirring occasionally, for another 2 minutes. Stir in the tomatoes and 2 cups water and simmer until all of the vegetables are softened, 5 to 8 minutes more.
Add the salt, pepper, and Parmesan cheese rind and stir. Add more water if needed for the desired consistency -- it should be very thick. Cover and simmer 30 minutes. Add the chickpeas and basil sprigs and cook until chickpeas are heated through, 5 to 6 minutes. Serve in bowls with the Parmesan cheese grated over top and freshly ground pepper, and drizzle with olive oil.