The halloumi sandwich is defined by a constraint the cheese itself imposes: it has to be eaten hot. Halloumi is a firm, salty, brined cheese with a high melting point, so it does not slump and run the way a Cheddar or an American slice does. Grilled or fried, it goes golden and develops a squeak and a slight chew; left to cool, it goes from squeaky to rubbery and dense, and the sandwich that was good five minutes ago becomes a tired one. Everything about the build is therefore organised around speed and warmth. The cheese is the whole sandwich, and the cheese has a clock on it.
The craft is the sear and the timing of assembly. Halloumi is sliced thick enough to take a hard, coloured face on a hot pan or grill without drying through, and it is built into the bread the moment it comes off the heat, not held. The bread is usually a soft roll or a sturdy bread that can take a hot, oily, salty filling without going limp, and it is the heat carrier as much as the wrapper. Because the cheese brings a lot of salt on its own, the counter is acid and freshness rather than more richness: lemon, a sharp chutney, rocket, something with bite that keeps the sandwich from reading as one salty register. Done right it is squeak, salt, and char against soft bread, eaten before it cools. Done slow it is the rubbery thing nobody finishes.
The variations almost all answer the same heat and salt problem from a different side. Halloumi with roast vegetables adds moisture and a Mediterranean reading and is its own build. Halloumi in a wrap with chilli and yoghurt borrows the kebab logic. A halloumi burger drops it into a bun with burger furniture; halloumi with harissa pushes the heat up against the salt. Each of those is its own sandwich and deserves its own article rather than being crowded in here.