In a medium bowl, crack the whole eggs and add the egg whites, Greek yogurt, sea salt, and black pepper.
Whisk the egg mixture vigorously until the yogurt is fully incorporated and the mixture is slightly frothy.
Roughly chop the smoked salmon into bite-sized pieces and finely mince the fresh chives.
Heat a non-stick skillet over medium-low heat and add the ghee, swirling to coat the pan.
Pour the egg mixture into the skillet and let it sit for 30 seconds until the edges begin to set.
Using a silicone spatula, gently push the eggs toward the center to create large, soft curds.
When the eggs are nearly cooked but still look slightly wet, fold in the smoked salmon and half of the chives.
Remove from heat immediately to prevent overcooking; the residual heat will finish the eggs.
Transfer to a plate and garnish with the remaining fresh chives before serving.