cheese is not confined

The way my school is set up, each class has two teachers. One is a foreign teacher (like me), and the other is a Korean teacher. The Korean teachers generally focus on English grammar, while the foreigners cover a range of areas: reading, pronunciation, comprehension, etc.

Often, when one of the Korean teachers is struggling to understand something to do with English, they will ask me what is right. Usually I can give them the answer and explain why it is; sometimes though it's not so easy. I can quickly identify what the right answer is, but explaining it is a lot more difficult.

For example, today one of the Koreans was struggling with this: the book she is teaching with had the sentence, "He is making a cake." Her problem was that she was teaching that "cake" isn't something that you count, and so she wondered if the book was wrong and should instead read, "He is making cake." I thought about it for a moment and told her that, well, both sound just fine. Confused, she asked if one could also say, "He is making a cheese." I told her no, that's not right, "He is making cheese" is the way to say that.

I mulled it over for a while, breaking it down for simplicity's sake. He has cake. He has a cake. Both can be used. He has cheese. He has a cheese. No.

I tried to explain that you could say, "He has a piece of cheese," and that would be fine, or you could use cheese as an adjective such has "He has a cheese sandwich," but you can't just say he has an ambiguous amount of cheese. A cheese could mean any amount, cheese is not confined*, whereas when you're talking about cake, everyone has a set idea of what a cake is.

The discussion ended as all of my discussions with Koreans about tricky aspects of the English language end. With me saying, "English is stupid like that," while the Korean smiles and says, "Okay, thank you."

So: I still don't entirely understand the reason why one cannot say, "He has a cheese." Do you?

*Upon rereading this post before publishing, I realized this statement had to be my title, supplanting my original title,"cake and cheese."