Preheat the oven to warm, or about 175 degrees. Place a cooling rack on top of a baking sheet.
In a large non stick skillet, melt one tablespoon of the butter. Add the onion and saute until beginning to soften, about 5 minutes. Stir in the green chilies and saute another 2 minutes. Remove the mixture to a bowl and set aside.
Add another sliver of butter to the skillet and melt, swirling it evenly around the pan. Add a tortilla, and spread cheese evenly over the whole thing, all the way to the edges.. Add a few spoonfuls of the green chile mixture to one half of the tortilla. Top with chopped chicken.
When the side without toppings is melty and starts to bubble, fold it over the toppings and use your spatula to press down so the cheesy sides melt together. If needed, give the quesadilla another 1-2 minutes to brown. Otherwise, remove to the cooling rack and place in a warm oven. Repeat with remaining tortillas and fillings.
Slice quesadillas into wedges with a knife or pizza cutter. Serve with sour cream, salsa or pico de gallo, or avocado lime dressing, for dipping.