Crack the eggs into a medium bowl, add a splash of milk, and season with kosher salt and freshly ground black pepper; whisk until blended.
Heat a nonstick skillet over medium-low and melt the unsalted butter until it coats the pan.
Pour the egg mixture into the skillet and stir gently and continuously with a spatula, cooking for about 2–3 minutes until mostly set but still soft.
Sprinkle the shredded cheddar over the eggs and continue stirring for about 1 minute more, until the cheese melts and the eggs are fluffy; then remove from the heat.
Transfer to a plate, top with chopped fresh parsley, and serve warm.