The piadina con verdure grigliate is the vegetable fold, and its logic is that the grill does to the filling what the testo does to the bread: it concentrates and chars rather than steams. Sliced courgette, aubergine, and peppers are cooked on a hot grill until they soften, lose their water, and pick up a bitter, smoky edge, then folded into a thin Romagnola round. The defining fact is that this only works when the vegetables are grilled dry and seasoned, not boiled or raw. Boiled vegetables would weep into the crumb and turn the fold to mush; raw ones would be loud and wet. Grilled and dressed, they have body, salt, and a charred sweetness that the plain warm bread has nothing of, and the round gives the loose pile something to hold it together in the hand.
The craft is driving the water out of the vegetables and binding them so the fold holds. Each vegetable is sliced to an even thickness and grilled hot enough to mark and soften without stewing, then drained and, ideally, left to give up its remaining steam before it goes anywhere near the bread, because the single thing that ruins this build is a wet filling soaking through a thin round. The vegetables are dressed with oil, salt, and often a little garlic or vinegar so they have seasoning of their own rather than relying on the neutral dough, and a soft cheese is frequently smeared in as much for bind as for flavour, gluing the loose slices to the crumb so the fold does not shed its filling on the first bite. The piadina is folded warm off the testo so it stays supple around the load; a sloppy build uses underdrained vegetables and a cool round and collapses into a soggy seam.
The variations are the cousins that change the bind or the vegetable rather than the method. There is the version held together with squacquerone or stracchino so the cheese carries the fat and salt, the one with grilled radicchio or other bitter leaves swapped in, and the leaner build dressed with oil alone and no cheese at all. Each of those is the same grilled-and-dried idea with one element changed, and each deserves its own article rather than being crowded in here.