In a large pot (6 to 8 quarts), heat oil over medium. Add onion and cook until soft, stirring, 2 to 3 minutes. Add garlic and red-pepper flakes; cook until fragrant, 1 minute. Add potatoes and broth; bring to a boil. Reduce heat; simmer until potatoes are tender, 10 to 15 minutes.
In a blender, puree half the soup. Return to pot; add kale and sausage. Simmer until kale is wilted, 10 to 15 minutes.