Render the pancetta. Heat olive oil (3 tablespoons) in a large Dutch oven over medium heat. Add pancetta (2 ounces) and cook until the fat renders and the pieces are golden and crisp, about 5–7 minutes. Spoon off excess fat if needed, leaving about 2 tablespoons in the pot.
Build the soffritto. Add onion (1), carrots (2), and celery (1 rib). Season lightly with salt and cook, stirring often, until vegetables are soft and lightly golden, about 6–8 minutes. Add garlic (1 tablespoon) and cook for 30 seconds, just until fragrant.
Deglaze. Pour in the white wine (1/2 cup), scraping up any browned bits. Simmer for 2–3 minutes, until reduced by about half.
Caramelize the tomato paste. Stir in tomato paste (2 tablespoons) and cook for 1–2 minutes, stirring constantly, until it darkens slightly and coats the vegetables.
Add tomatoes, broth, and aromatics. Add crushed tomatoes (1 can), bay leaves (2), rosemary (1 sprig), thyme (2 sprigs), Parmesan rind (1), and red pepper flakes (1/4 teaspoon) (if using). In a measuring cup or bowl, whisk the Better Than Bouillon (1 tablespoon) into a splash of the chicken broth until smooth, then add it to the pot along with the remaining chicken broth (8 cups). Bring to a gentle boil, then immediately reduce heat to low and simmer uncovered for 20 minutes.
Add beans and thicken. Mash 1/2 cup cannellini beans with a ladle of hot broth until smooth. Stir the mashed beans, remaining cannellini beans, and kidney beans (1 can) into the soup. Simmer for another 15 minutes, stirring occasionally.
Season the broth before the pasta. Stir in 1/2 teaspoon fine salt (this helps season the pasta as it cooks). Simmer 1 minute, then taste—remember you’ll add Parmesan later, so keep it lightly seasoned for now.
Cook the pasta. Stir in pasta (6 ounces) and simmer until just tender, 8–10 minutes, stirring to prevent sticking.
Finish and rest. Remove and discard bay leaves, herb stems, and Parmesan rind. Remove from heat and let the soup rest for 10 minutes to settle and thicken naturally.
Final seasoning. Stir in Parmesan cheese (1 cup). Season it with salt and freshly ground black pepper to taste. Add red wine vinegar, starting with 1 teaspoon, to brighten and balance.
Serve. Ladle into bowls and finish with a drizzle of olive oil, extra Parmesan, chopped parsley, and black pepper.