Melt butter in a microwave-safe mug for about 10–15 seconds; swirl to coat the sides.
Whisk in beaten eggs, milk, salt, and pepper until uniformly mixed and slightly frothy.
Microwave on high for 30–35 seconds, then stir, scraping the sides and bottom.
Heat another 20–30 seconds; stir again to break up large curds.
Microwave 15–25 seconds more until softly set and glossy. Fold in cheese and chives, then rest 30 seconds before serving.