Using an immersion blender on high, combine eggs and heavy cream until smooth, then set aside.
Roughly chop half of the smoked salmon into pieces, then set aside. Using the remaining smoked salmon, roll each slice lengthwise with uneven edge exposed, forming rosette shapes, set aside.
Melt butter in a nonstick skillet over medium heat and add eggs, stirring constantly and lifting off the bottom to avoid browning. When the eggs are about three-quarters cooked, turn off the heat—the residual heat in the pan will finish cooking the eggs—and fold in the chopped salmon and chives until eggs are slightly wet but cooked through.
Transfer contents of skillet to serving dish and top with smoked salmon rosettes, capers, and parsley. Serve immediately.