Can you add cheese to a veloute?

You don't need milk. There are two common ways to help prevent the sauce from splitting on you -- starch (used in mornay), and acid.

When Alton Brown did his "Good Eats: Reloaded", he admitted that his original recipe for fondue was a problem ... he had used a more acidic hard cidre, which meant it worked fine for him, but caused major problems for people trying it at home. His updated fondue recipe calls for both cornstarch and lemon juice.

This also typically means that "processed" cheeses won't break (as they typically contain a little bit of starch and acidic salts), and that pre-shredded cheese has less problems (as they coat it in starch to prevent it from clumping back together). So if someone develops a recipe that works for them, but you shred your own cheese, it might not work.

I suspect that adding cheese to veloute would work, but you want to work the cheese in slowly, over a moderate amount of heat, and you may wish to add a little bit of extra acid for insurance.

By coincidence, last night I made a soup involving stock, potatoes, and aged Irish cheddar cheese. While the recipe had milk, it was added after the cheese was completely melted. The cheddar melted and distributed through the soup easily and uniformly before I added the milk.

Based on this experiment, I'd say that veloute' with cheese should be possible, and that the only things required to get good dissolving of the cheese are lots of starch plus the right temperature (85-95C). Acidity is apparently not a requirement, nor is using processed cheese products, if you have enough starch.