Preheat a large non-stick skillet to medium-high heat. Lightly butter a single side of one tortilla with about ½ a teaspoon of butter and place it in the hot pan. Top half of the tortilla with about half of the cheese. Spread the onions across the cheese and top with chicken.
Sprinkle the rest of the cheese over the chicken, and fold the tortilla over on itself. Let it cook for 2-3 minutes. When the cheese has begun to melt and the bottom 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.