Counterpoise Query: Kebab-Curious

Kebab. Shawarma. Gyro. Doner. Understanding the cultural and geographic points of departure and distinction amongst the myriad names and phenotypes of the many tasty snacks of broadly Mediterranean extraction, consisting of meat wrapped inside flattened bread, would take possibly years of work to accomplish without oversight. Initial research on this topic did reveal some commendable efforts, such as this obsessive discourse, which delineates some seven different types of wrapped proteins, divided into two chief categories: on-the-spit, and on-the-grill.
Perhaps the task of creating a governing taxonomy that could capture the multitude of regional idiosyncrasies occurring within the broad range of the gyro-kebab-shish zone of origins that extends from northern Greece to the tip of the Arabian peninsula is a hopeless one, as if an immortal biologist were attempting to classify distinct animal species when viewing Darwinian evolution in geologic time. Indeed, the kebab at least has been described as one of the region's most ancient dishes.

The significance of the treat known generally as "kebab" in modern times is great, and nowhere is it greater than in urban areas of the former British Empire, where it has become perhaps the favorite late-night craving, in a multitude of forms as discussed on the website above, of the young drinking classes of these cities.
This leads us finally to our query: Can anyone explain how it is that some kebab shops, especially in Australia, have come to grilling already stuffed, folded kebabs? One of our contributors was smitten with this particular embellishment to her kebabs when she was down under (pronounced colloquially "ke-baabs", in an annoying, if not morbid, approximation of a lamb's vocalization) . Since then, she has been unsuccessful in locating a place stateside that does the griddle finishing move. Is this a technique as ancient as the kebab itself, or an ingenious recent innovation of the mostly Turkish immigrants who serve the stuff in Aussieland? Counterpoise wants the 411.