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Preheat the oven to 400°. Brush a large rimmed baking sheet with 1 tablespoon of the olive oil. Arrange 4 of the tortillas on the baking sheet.
In a medium bowl, toss the crabmeat with the cheese, cilantro, scallion and jalapeños and season with salt. Spoon the crab mixture over the tortillas and top with the remaining tortillas. Brush the tortilla tops with the remaining 1 tablespoon of olive oil.
Bake the quesadillas for about 12 minutes, flipping them halfway through, until the cheese is melted and the tortillas are lightly golden and crispy. Cut into quarters and serve.