In a small bowl, crack the eggs and whisk them together with a pinch of salt and pepper until well combined.
Heat a non-stick skillet over medium heat and add the butter. Once melted, pour in the whisked eggs. Cook, stirring gently, until the eggs are scrambled and just set, about 2-3 minutes. Remove from heat.
Place one tortilla in the skillet over low heat. Sprinkle half of the shredded cheese evenly over the tortilla.
Spoon the scrambled eggs over the cheese, then top with the remaining cheese. Place the second tortilla on top.
Cook for about 3-4 minutes, or until the bottom tortilla is golden brown and the cheese begins to melt. Carefully flip the quesadilla and cook for another 3-4 minutes until the other side is golden brown and the cheese is fully melted.
Remove from the skillet and let it cool for a minute before cutting into wedges. Serve with salsa or hot sauce if desired.