Finely chop the fresh chives and set half aside. Ensure the smoked salmon snippets are at room temperature. Crack the eggs into a bowl and add the milk or cream, along with some salt and freshly ground black pepper. Whisk everything vigorously with a whisk until light and evenly combined. Then stir the first half of the chives through the batter.
Heat a frying pan over medium-high heat and melt the unsalted butter. Pour the egg mixture into the pan once the foam from the butter has subsided. Leave the eggs untouched for twenty seconds so the edges can set. Push the set edges toward the center with a spatula and tilt the pan to let the still-liquid egg run to the edges. Repeat until the omelette is mostly set but the top still feels slightly moist.
Distribute the salmon snippets over one half of the omelette. Use the spatula to fold the empty half over the salmon. Gently press the omelette together and let it sit on the heat for another thirty seconds so the filling can warm through.
Gently slide the omelette onto a plate from the pan. Sprinkle the reserved chives on top and serve immediately.