Heat up a nonstick frying pan on a medium-low heat.
Place one tortilla in it, and top with a quarter of the cheese in an even layer.
Top with half the cabbage in an even layer.
Add another portion of the cheese on top of the cabbage, and top with another tortilla.
Press down lightly with a spatula to encourage the filling to meld together.
When the bottom tortilla is golden brown, and the cheese is melting, slide the quesadilla onto a plate, then flip it back into the pan.
Cook gently on the other side until both sides of the quesadilla are golden brown, and the cheese is thoroughly melted.
Place the quesadilla on a serving plate or board. Squeeze Japanese mayonnaise and okonomiyaki sauce over the quesadilla, then top with a mound of bonito flakes.