Crack the eggs into a medium bowl, add a splash of milk, a pinch of kosher salt and freshly ground black pepper, then whisk until combined.
Heat the unsalted butter in a nonstick skillet over medium-low heat until melted and slightly foamy.
Pour the egg mixture into the skillet and stir gently and continuously with a spatula, cooking over medium-low heat for about 2–3 minutes until soft curds form.
Sprinkle the shredded cheddar over the eggs, continue stirring for about 1 minute until the cheese melts and the eggs are fluffy, then remove the pan from the heat.
Transfer the scrambled eggs to a plate, top with the chopped parsley, and serve warm.
