How to Make It
Preheat the broiler and set a rack about 6 inches from the heat. Line a rimmed baking sheet with parchment paper and spread the tomatoes and poblanos on the sheet. Broil for 4 to 5 minutes, turning the tomatoes once, until softened and charred in spots. Transfer the tomatoes to a work surface. Broil the poblanos for 4 to 7 minutes longer, turning occasionally, until charred all over.
Let the tomatoes and poblanos cool. Coarsely chop the tomatoes; peel, seed and stem the chiles, then thinly slice them crosswise. Leave the broiler on.
Preheat a cast-iron griddle or large, heavy skillet. Brush the tortillas with olive oil. Toast the tortillas on the hot griddle, turning once, until browned in spots and crisp, 4 minutes.
In a bowl, toss together the asadero, Oaxaca and Manchego cheeses. Arrange 2 tortillas on each of 2 large rimmed baking sheets. Sprinkle the cheese mixture over the tortillas, covering the entire surface. Scatter the tomatoes and poblanos over the cheese. Top each tortilla with 2 tablespoons of the Cotija and 1/2 tablespoon of the oregano and season lightly with salt.
Broil the quesadillas one sheet at a time about 6 inches from the heat until the cheese is just melted, 3 to 4 minutes. Transfer the quesadillas to a work surface, cut into wedges and serve while you broil the second batch.
Asadero (also known as queso quesadilla) and Oaxaca are great for melting. Cotija is a white, dry, crumbly cheese. All of the cheeses are available at Mexican markets.