Heat the oil or butter in a large pot. Once melted, add the onion, garlic, and carrots and cook until softened, about 5 minutes.
Add broccoli. Cook for another 5 minutes, then season with salt, pepper, paprika, mustard and hot sauce. Add milk and broth. Boil on medium heat for 10 minutes.
Mix cornstarch and cold water. Pour into the pot and whisk until smooth. Let it boil to thicken.
Add the cheese and cook until the cheese melts. Adjust seasoning. Serve.
