I agree with the other members that 'special' is the better word to use here. A 'special' suggests that the dish is on offer just for the day, in addition to the regular items which are always on the menu. This is different from a 'speciality' ( or 'specialty', if you prefer) which might be a signature dish which a chef or restaurant is famous for. There is nothing at all wrong with the term 'goat's cheese', however - this is perfectly correct in British English. I'd always say 'goat's cheese'. When I google 'goat cheese', the only sites I can find are American ones.

Also we say GOAT cheese, not GOAT'S.

Yes it would generally be today's special. Also depending on what the dish is it may be better to say "made with goat cheese" assuming it's a dish with more ingredients vs a dish that is almost entirely goat cheese that has been changed in some way.

Yes. A "specialty" is a more like a special skill or product that is more-or-less permanently available. Thus pairing it with "today's" sounds odd. "This restaurant's specialty is goat cheese, they have been using it for 20 years". "Learning English is his specialty, although he is also good at Math."