Preheat a large non-stick skillet to medium-high heat. Lightly butter a single side of one tortilla with about a teaspoon of butter and place it in the hot pan.
Sprinkle 3 tablespoons cheese over one half of the tortilla, then top with the carnitas, another 3 tablespoons cheese, and the cilantro. Fold the tortilla over and cook for 2-3 minutes.
When the cheese has begun to melt and the bottom half of the tortilla is browning slightly, flip the folded quesadilla over to cook the other side. Continue to cook until all the cheese is melted and the tortilla is crisp around the edges.
Using a spatula, transfer the quesadilla back to a cutting board and cut into wedges while it is hot and serve warm. Repeat as desired.