1. Heat the oil in a large stockpot or Dutch oven over medium heat until shimmering. Add the onion and cook, stirring frequently, until softened, about 5 minutes. Add the garlic and sauté until fragrant, about 30 seconds.
2. Add 3 cups stock, the potatoes, and 1/2 teaspoon salt. Increase the heat to medium-high and bring to a boil. Reduce the heat and simmer until the potatoes are tender, about 15 minutes. Remove the pot from the heat and mash the potatoes in the liquid with a potato masher until no large chunks remain and the potatoes thicken the soup slightly.
3. Return the pot to medium-high heat. Add the remaining 3 cups stock, sausage, and savory and bring to a boil. Reduce the heat to medium-low, cover, and simmer to blend the flavors, about 15 minutes.
4. Remove and discard the savory. Stir in the kale and simmer until just tender, about 5 minutes. Adjust the seasonings with salt and pepper to taste. Serve immediately, drizzling each portion with olive oil, if desired.