Heat a large stock pot over medium-high heat. Add the butter and olive oil to the pan and let it melt.
Once the butter is melted, add in the celery, onion, carrots, garlic and prosciutto. Cook for 5 minutes, stirring occasionally, until vegetables are softened.
Add the green beans, cabbage and zucchini to the pot. Cook for an additional 5 minutes, stirring occasionally.
Pour in the chicken stock, dried parsley, dried basil, tomatoes and season with salt and pepper to taste. Stir to combine all ingredients together. Bring soup to a boil, then reduce heat and let simmer for 20 minutes.
Add the potatoes, garbanzo beans, kidney beans, and pecorino romano rind to the pot and let simmer for an additional 20 minutes or until the potatoes are tender.
Remove bay leaf from soup, then add grated Pecorino Romano cheese. Stir to combine and let the cheese melt.
Serve individual bowls of soup with extra Pecorino Romano cheese on top if desired.