Pan stain after roasting cheese A couple of weeks ago I bought this "grill and pan cheese" in a supermarket and I distinctly remember to have chosen this one because it didn't contain any weird chemicals... It seemed to me "only cheese" and nothing else.
It came in a parallelepiped form (just like feta would... and looked just like that) and both grilled and pan-roasted, its consistency was a bit chewy (to the point that I would hear squeaking every time that I put a piece in my mouth).  

I'm explaining this whole thing to help you better understand what I am talking about, so you can maybe have a better opinion on the fact.

When I roasted the squeaky thing in the pan to the point that it was brown on both sides, I noticed that my pan was stained. After the meal I rinsed it (when it was cool), and put it in the dishwasher. After having it washed, the rectangular brownish stain remained (and I could also not remove it washing the pan by hand). Notice that the pan is a ceramic "non-stick" one.

What do you think caused the stain? How can I clean the pan?

EDIT:

Just a follow up... After all this time my pan never failed me. Nothing else sticked to it or caused nasty stains. The one of the cheese remained though.
The most likely explanation is that your pan starts failing. When ceramic pans fail, they do so by getting gummed up with food buildup. 

I am a bit surprised to see it first occur with cheese, but maybe you dipped it in batter before frying? Also, the herbs on it could have been mixed with starch before being applied to the cheese, this is done to prevent caking and mold. Typically, the worst combination for these pans is starch with fat. On the other hand, I have no proof that it can't happen with cheese, it's just that I haven't heard of it being the first food to start sticking. 

Once this happens, you can't really clean it. A lye solution will restore it for a short time, but I don't know the physics of the pan failure, so I have no idea if it is beneficial, or on the contrary, if it speeds up the degradation process. 

Just continue using the pan, and once it becomes too non-nonstick, decide if you want to keep it that way and fry with it as with a standard pan (it behaves much like a stainless steel pan afterwards) or throw it out. For me, it depends on the pan heating quality, if it is a good pan with thick bottom and even heating, it continues doing a good job after the coating fails.