In a large skillet, heat the butter over medium-low heat until melted and starting to bubble.
While the butter is melting, whisk together the eggs, cream, salt, and pepper in a mixing bowl until creamy and smooth. Pour into heated skillet.
Sprinkle the cheese on top and let sit in the skillet, undisturbed, for about 30 seconds to a minute (when you start to see the eggs firm up and become slightly opaque around the edges).
Using a silicone spatula, fold the edges of the eggs towards the center. Continue scrambling in this way, scraping melted cheese into the eggs, until the eggs are cooked to your preferred firmness (time will vary depending on if you like more wet or dry eggs). About halfway through, the cheese will get pretty stringy but will come together more with the eggs towards the end.
Serve hot with a sprinkling of freshly chopped chives.