Heat 2 tablespoons olive oil in large stock pot over medium-high heat. Add pancetta; cook and stir until it begins to brown, 2 to 3 minutes. Stir in onions and celery; cook and stir until onions start to turn translucent, about 3 minutes. Stir in minced garlic and cook for another minute.
Pour chicken broth, water, and plum tomatoes into the pancetta and onion mixture. Bring to a simmer.
Stir cranberry beans, cabbage, garbanzo beans, red pepper flakes, Italian seasoning, and 2 teaspoons salt into broth mixture. Bring to a simmer and cook until cranberry beans are tender, adding more water as needed if the soup becomes too thick, about 45 minutes.
Stir in Swiss chard and simmer until softened, about 15 minutes. Season with salt and black pepper to taste.
Stir in pasta and increase heat to medium-high and simmer until pasta is tender, about 15 minutes. Ladle into bowls and top with extra virgin olive oil, Parmigiano-Reggiano cheese, and Italian parsley.