In a medium bowl, whisk together eggs, milk, salt, and pepper until just combined and no streaks remain. Do not overbeat - you want fluffy curds, not foam.

Heat 1 teaspoon butter in a 10-inch nonstick skillet over medium-low heat. Pour in the egg mixture and let sit for 10 seconds until edges just set. Gently push and fold the eggs with a spatula, forming large soft curds. Cook until eggs are mostly set but still slightly glossy, about 2 to 3 minutes total. Transfer to a plate immediately to stop cooking.

Wipe out the skillet and return to medium heat. Place one tortilla in the dry skillet. Sprinkle 1/3 cup cheese over half the tortilla, top with one-quarter of the scrambled eggs, then another 1/3 cup cheese. Fold the empty half over to create a half-moon.

Cook until the bottom is golden brown with crisp spots, 2 to 3 minutes. Carefully flip and cook the second side until cheese is fully melted and tortilla is crisp, another 2 minutes. Transfer to a cutting board.

Repeat with remaining tortillas and filling. Let each quesadilla rest for 1 minute, then cut into wedges. Serve warm with salsa and sour cream if desired.