In a large soup pot, heat the vegetable oil over medium-high heat until hot but not smoking. Add the sausage and fry, stirring occasionally, until browned, 5 to 7 minutes. Use a slotted spoon or tongs to transfer the sausage to a medium bowl and set aside.
Add the butter to the pot and melt over medium heat. Add the onion, potatoes, celery, and carrots and cook, stirring occasionally, until the onion is translucent and the vegetables are slightly softened, about 10 minutes. Stir in the flour and cook, stirring constantly, for 1 minute.
Add the broth, sausage, salt to taste, and the pepper. Increase the heat to medium-high and bring to a boil. Reduce the heat and simmer until the vegetables are tender and the flavors are combined, about 30 minutes.
Stir in the kale and simmer until wilted, about 5 minutes. Ladle the soup into bowls. Serve hot.
The soup is best served soon after the kale is added. Store leftovers in a covered container for up to 2 days.