Crack all six eggs into a medium bowl and pour in the heavy cream. Drop in the small pieces of cream cheese along with a pinch of salt and the black pepper. Whisk everything together until it's mostly smooth; a few small lumps of cream cheese are totally fine and will melt in the pan.
Set a nonstick pan over medium-low heat and add the butter. Let it melt slowly rather than cranking the heat, since low and slow is the secret to eggs that stay soft instead of turning tough or dry. Once the butter is fully melted and just starting to foam, you're ready for the eggs.
Pour the whisked egg mixture into the warm pan. Use a spatula to stir slowly and continuously, pushing the eggs from the edges toward the center. This gentle, constant motion is what creates those soft, creamy curds instead of dry, overcooked chunks.
Once the eggs are nearly set but still slightly glossy, usually around 2 to 3 minutes, add the chopped smoked salmon. Gently fold it through the eggs so it warms through without breaking apart too much. This should only take another 30 to 40 seconds.
Transfer the eggs to a serving dish right away, since they'll keep cooking a bit from residual heat if left in the pan. Scatter the chopped chives or scallions on top for a pop of color and a fresh, mild bite. Serve immediately while everything is still warm and creamy.