Coat the bottom of a large pot with the oil and place it over medium heat. Give the oil a minute to heat up, then add the onion. Cook the onion for about 5 minutes, stirring occasionally, until it becomes soft and translucent. Stir in the garlic and cook it for about a minute longer, until very fragrant.
Stir in the broth and potatoes. Raise the heat and bring the liquid to a boil, then lower the heat and let it simmer for about 15 minutes, until the potatoes are easily pierced with a fork.
While the soup simmers, remove the stems from your kale and collards, then slice the leaves very thinly.
Use a slotted spoon to remove about half of the potatoes from the soup, placing them into a heatproof bowl. Use a fork or potato masher to mash the potatoes until creamy.
Return the mashed potatoes to the pot, stirring until they're well incorporated, then add the kale and collard greens to the pot. Raise the heat and bring the soup back to a boil, then lower the heat and let it simmer for at least 5 minutes, or until the greens have reached your desired level of tenderness.
While the greens simmer, place a medium skillet over medium-high heat and add the sliced sausage links. Cook the sausage slices for about 5 minutes, stirring occasionally, until they're browned and a bit crisp.
Stir the sausage into the soup, then add the red wine vinegar and season it with salt and pepper to taste.
Ladle the soup into bowls and serve.