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Note: The prep time includes over an hour for properly caramelizing onions, which can be done up to several weeks in advance.
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Freeze caramelized onions until ready to thaw and use.
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To caramelize the onions, heat butter and half of the olive oil over medium high in a large skillet until butter melts.
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Add sliced onions, sprinkle with half of the salt, and stir constantly for 15 minutes.
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Reduce heat to medium low and stir every 5 to 10 minutes until onions caramelize, which will take anywhere from 60 to 90 minutes, depending on the heat used.
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Onions may not darken, but theyre ready when they have a jam-like consistency and taste quite sweet.
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Store in the refrigerator or freezer if not using immediately.
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Wipe out the skillet and heat remaining half of the olive oil over medium.
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When shimmering, add sliced mushrooms.
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Stir infrequently as they cook and eventually release their liquid, which will evaporate.
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This will take about 10 minutes total.
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Sprinkle lightly with salt and set aside.
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For each quesadilla, heat 1 1/2 teaspoons vegetable oil over medium high in a non-stick skillet thats large enough to hold 1 tortilla.
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Lay a tortilla in the oil, and cook on one side only, about 45 seconds.
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Remove it from the skillet and place oiled side down on 2 layers of paper towels.
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Repeat with a second tortilla.
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Fry remaining tortillas the same way, two at a time, in additional oil.
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Carefully wipe out the skillet with a paper towel, leaving just a trace of oil.
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With the quesadilla ingredients accessible, place one tortilla, oil side down, in the skillet, and turn heat to medium.
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Layer a small handful of cheese (1/3 to 1/2 cup), followed by a fourth of the caramelized onions, mushrooms, and red pepper.
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Sprinkle with black pepper, top with more cheese, and lay second tortilla on top, oiled side up.
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Weigh down the quesadilla with a small, heavy pan, or using the back of your spatula, and cook about 2 minutes (it will be sizzling gently).
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To flip, pick up quesadilla on a large spatula, and use all your courage to give it a brave toss so it lands on the other side.
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Cook another 2 minutes, until the second side is crisped.
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Then remove it from the skillet and repeat the process to make the remaining quesadillas.
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Hold finished quesadillas in a warm oven until serving.
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Cool slightly before cutting each into quarters and serving with salsa or dipping sauce of choice.