Heat the olive oil in a medium skillet. Add the onion, season with salt and pepper and cook over moderate heat, stirring, until the onions are beginning to brown, about 8 minutes. Add the water and cook until it has evaporated and the onion is very tender and lightly caramelized, about 10 minutes. Let cool.
Dice the apple into 1/2-inch chunks and toss with the lemon juice.
Arrange the tortillas on a work surface and top each tortilla with cheese, apple and onion. Fold each tortilla in half to make 4 quesadillas.
In a large skillet, heat the vegetable oil. Add the quesadillas and cook over moderately high heat until golden and crisp, about 1 to 2 minutes per side. Add more oil to the skillet if necessary. Serve immediately.
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