There are hundreds of versions of this classic recipe. It's a favorite in Spain, Portugal and some of the Azorean Islands (off the lower coast of Portugal). It's quite easy to prepare, and it's delicious and features a full, robust flavor palate. In Spain, it would be prepared with Spanish Chorizo, and in Portugal and the Azores it would be prepared with linguica. The addition of the tomatoes (a popular ingredient in Spanish and Azorean cuisines, in particular) makes this version a little different from other Spanish/Portuguese Kale Soup recipes that you'll find on most sites.
Place the sausage in a large pot over medium-low heat and cook until it begins to render its fat, about 2 minutes. Add the onion and cook for 2 minutes. Add the garlic and potatoes and cook for 2 minutes. Add the kale and cook, stirring constantly, for 2 minutes longer.
Stir in the chicken broth, vinegar and salt. Bring to a boil, reduce heat, cover and simmer for 1 hour. Season with pepper. Stir in the tomatoes and cook, uncovered, for 15 minutes. Divide among 4 bowls and serve.