Heat the extra-virgin olive oil in a large heavy pot over medium heat until shimmering, 1–2 minutes.
Add the yellow onion, carrot, and celery; cook, stirring occasionally, until softened and lightly golden, 7–8 minutes.
Stir in the garlic and tomato paste; cook until fragrant and the paste darkens in color, 1–2 minutes.
Add the canned whole peeled tomatoes with their juices; cook, stirring, until slightly thickened, 3–4 minutes.
Pour in the vegetable broth and add the bay leaf, Parmigiano-Reggiano rind, potato, and green beans; bring to a boil over high heat, 3–4 minutes, then reduce to a gentle simmer and cook 10 minutes.
Stir in the zucchini and savoy cabbage; simmer until all the vegetables are tender but not mushy, 10–12 minutes.
Add the canned cannellini beans and ditalini pasta; simmer, stirring occasionally, until the pasta is al dente, 8–10 minutes.
Fish out and discard the aromatics added earlier; season the pot with the kosher salt and black pepper, stir in the flat-leaf parsley, and let the soup rest off the heat for 5 minutes to meld flavors.
Ladle into bowls and top with Parmigiano-Reggiano cheese, finely grated (for serving). Serve hot.