Heat half of the butter in a nonstick skillet over medium heat. Add the eggs and cook, stirring frequently, until just set, about two minutes. Transfer the eggs to a bowl, and wipe out the skillet.
Add the remaining butter to the skillet. Place one of the tortillas in the skillet, and sprinkle with half of the cheese. Add the eggs in an even layer, then sprinkle with the remaining cheese. Spread the refried beans over the second tortilla, and place, beans-side down, on top of the cheese. Cook about two minutes per side, until the tortillas are both crispy and the cheese is melted.
Transfer the quesadilla to a cutting board. Let it cool one minute before slicing and serving.