Crack the eggs into a small bowl, add the milk, salt, and freshly cracked black pepper, then whisk until blended and slightly frothy.
Heat a small nonstick skillet over medium-low heat until warm; adjust so it is not too hot (eggs should not fry) and not too cool (eggs may stick).
Add the butter to the pan and let it melt and coat the surface.
Pour the whisked eggs into the skillet and, using a spatula, continuously stir and fold the eggs, scraping the bottom, until they form soft curds and reach your desired doneness.
Remove the pan from the heat immediately when the eggs are cooked to your liking; the residual heat will finish them.
Transfer the eggs to a plate and top with the shredded cheddar and chopped chives, then serve.
