In a medium bowl, whisk together the eggs, heavy cream, salt, and black pepper until well combined and slightly frothy.
Heat a non-stick skillet over medium-low heat and add the butter. Allow it to melt and coat the bottom of the pan.
Pour the egg mixture into the skillet. Let it sit undisturbed for about 30 seconds until the edges start to set.
Gently stir the eggs with a spatula, pushing them from the edges toward the center. Continue to cook, stirring occasionally, until the eggs are softly set but still slightly runny. This should take about 4-5 minutes.
Remove the skillet from the heat and let the residual heat finish cooking the eggs. Stir in the chopped chives just before serving.