Arrange all your ingredients so that they'll be easy to add to the quesadilla. Lay several layers of paper towel on a work surface, and have a rack and tongs ready.
Heat a large skillet (just a bit larger than the tortilla) until it's very hot, add the oil, and when it's very hot (but not quite smoking), lay the tortilla in the oil. Cook until the underside is slightly blistered and light brown, about 20 seconds. Lift the tortilla with the tongs, hold it over the pan to let most of the oil drip back in (if you like, lay it on the paper towels for a second to blot even more oil). Put the tortilla back in the pan, uncooked side down.
Quickly distribute the filling ingredients evenly over the tortilla. It will puff up in places, but that's fine. When the second side of the tortilla is browned and crisp but still pliable, fold the tortilla in half with the tongs, transfer it to the paper towels, blot for a few seconds, and then transfer to the cooling rack while you make the next. Cut into wedges and serve immediately, with more salsa if you like.