Crack the eggs into a bowl, add the coconut milk and a pinch of salt, then whisk until combined and slightly frothy.
Mix the crème fraîche with most of the chopped chives (reserve some for garnish) and season lightly; set aside.
Slice the red onion very thinly and finely slice the spring onion; set aside.
Heat a splash of olive oil and a knob of butter in a non-stick frying pan over medium heat until the butter foams.
Pour the egg mixture into the pan, scatter over the sliced spring onion, and cook gently until the edges are set and the top is mostly firm, about 3–5 minutes.
If desired, place the pan briefly under a very hot grill to set the top and help the omelette puff (watch closely).
Remove the omelette to a plate, dollop over the chive crème fraîche, arrange the smoked salmon ribbons on top, and scatter the red onion and remaining chopped chives.
Finish with a grind of black pepper and a squeeze of lemon, then serve immediately.
