Heat 3/4 tablespoon olive oil in a large saucepan over medium heat. Sauté the minced onion until softened, about 3 minutes. Add the garlic and cook for 1 minute.
Add the sliced potatoes to the pot, stir well, and cover with the water or chicken stock. Bring to a boil, then reduce the heat and simmer for about 20 minutes, or until the potatoes are very tender.
In a separate skillet, cook the sausage slices over medium heat until browned and some fat has rendered, about 8–10 minutes. Remove from heat and set aside.
Use an immersion blender to purée the soup until smooth. If using a regular blender, work in batches and be cautious with hot liquids.
Return the puréed soup to medium heat. Stir in the browned sausage and julienned kale. Simmer for 5–7 minutes until the greens are tender and vibrant.
Adjust seasoning with salt and black pepper. Serve hot with crusty bread.
