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1
Preheat the broiler and set a rack about 6 inches from the heat.
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2
Line a rimmed baking sheet with parchment paper and spread the tomatoes and poblanos on the sheet.
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3
Broil for 4 to 5 minutes, turning the tomatoes once, until softened and charred in spots.
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4
Transfer the tomatoes to a work surface.
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5
Broil the poblanos for 4 to 7 minutes longer, turning occasionally, until charred all over.
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6
Let the tomatoes and poblanos cool.
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7
Coarsely chop the tomatoes; peel, seed and stem the chiles, then thinly slice them crosswise.
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8
Leave the broiler on.
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9
Preheat a cast-iron griddle or large, heavy skillet.
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10
Brush the tortillas with olive oil.
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11
Toast the tortillas on the hot griddle, turning once, until browned in spots and crisp, 4 minutes.
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12
In a bowl, toss together the asadero, Oaxaca and Manchego cheeses.
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13
Arrange 2 tortillas on each of 2 large rimmed baking sheets.
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14
Sprinkle the cheese mixture over the tortillas, covering the entire surface.
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15
Scatter the tomatoes and poblanos over the cheese.
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16
Top each tortilla with 2 tablespoons of the Cotija and 1/2 tablespoon of the oregano and season lightly with salt.
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17
Broil the quesadillas one sheet at a time about 6 inches from the heat until the cheese is just melted, 3 to 4 minutes.
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18
Transfer the quesadillas to a work surface, cut into wedges and serve while you broil the second batch.