Place the butter in a 10-inch nonstick skillet. Heat it over medium heat until melted.
Add the egg halves. Fry them until golden brown, about 3 minutes per side. If the butter browns too much, reduce the heat to medium-low and keep cooking.
Season the eggs with salt and pepper and serve.
Video
Notes
You want the yolks to be fully cooked. So, if you use my tutorial for how to boil eggs, keep them in the hot water for at least 12 minutes.
You can keep the leftovers in a sealed container in the fridge for 3-4 days. But if you've used hard-boiled eggs that have already spent a few days in the fridge, take that into account. Reheat the leftovers in the microwave at 50% power, or enjoy them cold.
I don't recommend freezing the leftovers. Cooked eggs become dry and rubbery when frozen and thawed.