Slice sausage into 1/4-inch rounds. Heat oil in a pot over medium-high heat. Add sausage and saute until sausage begins to turn brown and renders some fat, 3 to 5 minutes. Transfer with a slotted spoon to a dish. Reduce heat to medium-low; add onions and pinch of salt salt. Stir and cook until softened, 4 to 5 minutes.

Transfer potatoes to pot. Sprinkle in 2 teaspoons salt. Pour in chicken broth. Increase heat to high and bring to a simmer. Reduce heat to medium-low; simmer until potatoes are tender, 8 to 10 minutes.

When potatoes are tender, carefully mash them in the broth with a potato masher until broken up or completely mashed. Add kale, a handful at a time. Transfer browned sausage to soup. Bring to a simmer over medium-high heat. Reduce heat to medium-low and simmer until greens are tender, about 45 minutes.