Heat 2 tablespoons olive oil in a stockpot over medium-high heat. Add pancetta; cook, stirring, until it begins to brown, 2 to 3 minutes. Stir in onion and celery; cook until onion turns translucent, about 3 minutes. Stir in minced garlic and cook 1 minute more.

Add chicken broth, plum tomatoes, and water. Bring to a boil, then reduce to a simmer.

Add cabbage, garbanzo beans, cranberry beans, 2 teaspoons salt, red pepper flakes, and Italian seasonings. 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; 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.