How to Make It
In a nonreactive bowl stir together the zucchini, onion, lime juice, chile, and salt. Cover the bowl and let sit at room temperature for 2-3 hours. When the red onions are bright pink, drain and squeeze out as much of the pickling liquid as possible and set aside.
In a non-stick skillet, heat 1/2 teaspoon of the oil. Place one tortilla in the skillet and sprinkle 1/4 cup of grated cheese evenly over the top. Scatter a 1/4 cup of the pickled zucchini and onion mixture over the first layer of cheese and then sprinkle with another 1/4 cup cheese. Top the quesadilla with the second tortilla and cook over moderately high-heat until crisp on the bottom, about 3 minutes. Carefully flip it over and cook on the second side for another 2-3 minutes, until cheese is melted and both sides are golden. Repeat with the remaining oil to make the second quesadilla. Slice into wedges, transfer to plates and top with the leftover salsa. Serve right away.