First, place your poblano chiles directly over a gas flame or under your oven's broiler. Turn them every few minutes until the skin is blackened and blistered all over. This charring is what creates that essential smoky flavor for the dish.
Once charred, transfer the hot chiles to a bowl and cover it tightly with plastic wrap. Let them steam for about 5-10 minutes. This process loosens the skin, making it much easier to peel away, revealing the tender pepper underneath.
After peeling the chiles, remove the stems and seeds, and slice the peppers into strips. In a separate bowl, combine the shredded Oaxaca or mozzarella cheese, cream cheese, garlic powder, cumin, salt, and pepper. Mix everything until it forms a creamy, cohesive filling.
Lay one large flour tortilla on a flat surface. Spread the cheese mixture evenly over the entire tortilla. Arrange the roasted poblano strips on top of the cheese, ensuring they are well-distributed so you get smoky pepper in every single bite.
Place another tortilla on top to close it. Heat olive oil or butter in a large skillet over medium heat. Carefully place the assembled quesadilla in the hot pan. Cook for 3-4 minutes on each side, until the tortilla is golden brown and crispy.
Remove the quesadilla from the skillet and let it rest for a minute. This allows the cheese to set slightly, making it easier to slice without the filling spilling out. Use a pizza cutter or a sharp knife to cut it into wedges and serve immediately.