Heat a large soup pot over medium-high heat until hot. Add the olive oil and the sausage rounds; cook and stir until sausage is browned. Remove sausage from pan.
Pour off all but 2 tablespoons of olive oil from pan (or add more olive oil if you have particularly lean sausage); add onion and garlic; cook and stir until the onion is translucent and the garlic is lightly toasted (taking care not to burn it), about 5 minutes.
Add 2 quarts water, potatoes, salt and pepper; bring to boil. Lower heat and simmer until potatoes are soft, 15 to 20 minutes. Mash some of the potatoes to add texture to the soup.
Add kale and cook until tender, about 5 minutes.
Return browned sausage to soup and stir. Season to taste with salt and freshly ground black pepper. Serve each bowl with a generous drizzle of extra virgin olive oil.