Brown sausage in a large pot with olive oil, breaking it into bite-sized pieces; remove excess fat if needed.
Add onion and cook until translucent, about 4 minutes; stir in garlic and oregano for 1 minute.
Add potatoes, broth, water, and beans; bring to a simmer and cook until potatoes are tender, about 12–15 minutes.
Stir in kale and cook until wilted, about 3 minutes; return to a gentle simmer.
Lower heat and stir in half-and-half; season with salt and pepper. Heat through but do not boil. Serve warm.