Remove all seeds and juice from the large tomato (or both small tomatoes) to prevent soaking the lavash.
Cut the remaining tomato flesh into small cubes.
Cut the cucumber into cubes or strips and add to the tomatoes.
Thinly slice the salad onion and add to the bowl. Finely chop the fresh greens (removing stems) and add to the bowl.
Cut or grate the 120 g of cheese into small pieces.
Mix all the filling ingredients together. Do not add salt—the cheese and teriyaki sauce provide enough seasoning.
Divide the lavash sheets into manageable pieces for wrapping.
Spread adjika on each lavash piece according to taste and heat level (use less if very spicy, more if milder).
Place the filling in the center of each adjika-spread lavash piece.
Wrap each lavash piece into an envelope shape, folding the edges to contain the filling.
Lightly brush a grill pan or frying pan with vegetable oil. Arrange the wrapped lavash envelopes in the pan.
Press a heavy lid or another pan on top and fry over medium heat for about 3-4 minutes per side until golden brown (or with brown stripes if using a grill pan) and the cheese melts inside.
Drizzle teriyaki sauce on the lavash envelopes just before serving.