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 eggs and scramble until fully cooked, about 2-3 minutes. Remove from heat.
Lay one tortilla flat in the skillet. 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 is starting to melt. Carefully flip the quesadilla using a spatula and cook for an additional 3-4 minutes on the other side.
Once both sides are golden and the cheese is melted, remove the quesadilla from the skillet and let it cool for a minute. Cut into wedges and serve with sour cream and salsa.