Ka'ak b'Jibneh (كعك بالجبنة) in this form is the sesame street bread filled specifically with akkawi or a similar brined white cheese. The angle narrows from cheese in general to the one cheese that defines the classic cart version, and akkawi brings a particular problem: it is a brined cheese that runs hard and saline straight from the tub and, untreated, can be both too salty and too firm to read as a soft filling. So the build is really about taming one ingredient and letting the bread's faint sweetness and toasted sesame do the balancing the cheese cannot do for itself.
The construction is short and the craft is in preparing the cheese. Akkawi is soaked or rinsed, often in changes of water, to pull the salt down and soften it before it ever meets the bread. The ka'ak is warmed so the sesame crust re-crisps and the crumb loosens, split along its wider belly, and the tempered cheese is sliced or grated in, sometimes blended with a milder melting cheese for a cleaner, stretchier result. A brief reheat softens the cheese and sets the crust. Tomato, mint, or a dusting of za'atar and oil are common to add acid and aroma against whatever salt remains. The judgment calls are the soak and the heat. Underprepared akkawi makes the sandwich a salty, squeaky brick; over-soaked, it goes bland and watery and weeps into the crumb. A good version has a soft, gently salty cheese with a short pull, a crisp sesame shell, and enough brightness to keep it honest; a poor one is either a salt slap or a soggy, flavorless pocket around a stale loaf.
It varies by how the akkawi is treated and what joins it. Some carts use it nearly straight for a sharper, saltier bite; others soak it long and cut it heavily with mozzarella-style cheese for pull and mildness, and many add nigella seed, mint, tomato, or za'atar. Within the ka'ak family it is the specific akkawi reading of the broader cheese filling, sitting beside the general soft-cheese version, the firm halloumi form that browns instead of melting, and the egg and labneh fillings. It is essentially the akkawi cheese sandwich the Lebanese table already knows, built into the sesame loaf and judged on whether the brine was brought to heel before the bread ever got warm.