I spent two weeks in Istanbul sampling as much local cuisine as I could, but with a city so vast you will need a year to try it all! But I must say that personally, my favorite gozleme (which I like to call a Turkish grilled cheese) was at the Gülhane Sur Cafe just a few minutes walk from Tokapi Palace in the Old City area of Istanbul. I must have visited this lovely location a half-dozen times during my stay, always greeted by the friendly and animated owner, Duman. You can also enjoy hookah and tea, or nargile and çay, on one of the many outdoor couches that offer a great people watching spot.
My suggestion: after a day touring the mosques and museums of Sultanahmet, come here to rest your feet and sample the crispy golden gozleme made by Duman's sister!