What you need to do is bind the cheese to the milk.
My sister accidentally discovered the perfect way to do this with mayonnaise.
For a sauce, use a 1:1 ratio of hard cheese and milk. First put the cheese in a pan and turn on a low heat. When the cheese softens, add the milk. Then add small amounts of mayonnaise and stir until the cheese fully binds with the milk.
This mix can be made into a soup base by adding milk. Alternatively, it can be cooled as used as a spread. You may season it how you like; I'm fond of basil and thyme.