Heat oil in a saucepan over medium-high heat. Add onion and cook until onion has released its water and begins to brown, about 15 minutes. Stir in flour and cook 1–2 minutes. Add potatoes and paprika; cook, stirring to coat potatoes, 1 minute. Stir in broth and Worcestershire. Bring to a boil, reduce heat to medium-low, and cook, partially covered, until potatoes are soft, 10–15 minutes. Coarsely mash potatoes with the back of a wooden spoon, and stir in cheese and milk until cheese is melted, 2 minutes. Remove soup from heat; season with salt and pepper.