Crack the cold eggs into a small saucepan or nonstick frying pan and add the butter.
Place the pan over high heat and begin stirring continuously with a rubber spatula, scraping the bottom and sides; do not whisk.
After about 30 seconds of stirring on the heat, remove the pan from the heat but continue stirring for 10 seconds.
Return the pan to the heat and continue the on-off stirring cycle (on the heat, then off while stirring) until the eggs are nearly cooked, repeating for a total of about 3 minutes of cooking time.
When the eggs are almost set but still slightly runny, remove the pan from the heat and stir in the crème fraîche and a pinch of salt and pepper until combined and creamy.
Transfer immediately to plates, garnish with chopped chives, and serve right away.
