Heat the broiler and place poblano on a rimmed metal sheet 6 inches from the heat. Broil for 8 to 10 minutes, turning until blistered, then cover in a bowl for 10 minutes, peel, seed and finely chop; resting is included in prep time.
Heat 1 tablespoon oil in a large skillet over medium-high heat. Cook mushrooms and onion for 7 to 8 minutes until their moisture evaporates and they begin to brown.
Add beef and cook for 7 to 8 minutes, breaking it into small crumbles, until it reaches 160°F. Drain excess fat and stir in cumin, ancho chile, salt and chopped poblano for 1 minute, then remove from the heat and stir in lime juice.
Divide half the Oaxaca over 8 tortillas. Spread one-eighth of the beef filling over each, add the remaining cheese and close with the other 8 tortillas to make 8 round quesadillas.
Heat 1 1/2 teaspoons of the remaining oil in a clean large skillet over medium heat. Cook 2 quesadillas for 2 to 3 minutes per side until golden, crisp and 165°F in the centers. Repeat in 3 more batches, using 1 1/2 teaspoons oil per batch.
Rest each batch for 2 minutes before cutting each quesadilla into 4 wedges. Serve 2 whole quesadillas, or 8 wedges, per person.