Heat the oil in a skillet over medium and cook the onion and poblano 5 to 6 minutes, until soft and no longer releasing steam. Stir in the cumin, salt and the shredded chicken, then take the pan off the heat.
Wipe the skillet out and set it over medium-low. A quesadilla needs a pan cool enough that the cheese melts before the tortilla colours.
Lay one tortilla in the dry pan. Scatter a quarter of the cheese over it, keeping a clear half inch at the rim, then a quarter of the filling, then a little more cheese on top.
Cover with a second tortilla and press it down with a spatula.
Cook 3 to 4 minutes, until the underside is patched golden brown and the cheese at the edge has started to run.
Slide the spatula fully underneath, flip in one motion, and cook another 2 to 3 minutes.
Lift it onto a board and wait 1 minute before cutting. Cut too soon and the molten cheese runs out of the wedges.
Repeat with the remaining tortillas. Cut each into 4 wedges and serve.