Crack the eggs into a bowl. Add the parsley and the cold water. Season with salt and pepper and use a fork to beat the eggs for barely four seconds to break the yolks.
Place a 15–24 cm frying pan (ideally non-stick) over medium-high heat. Add half the butter (or oil) and when it begins to sizzle pour in half the eggs. Swirl it around a bit and use a spoon or spatula to bring the set outer egg into the centre until it all begins to set.
Lay half the salmon and half the cheese across the centre and leave for another five seconds. Carefully fold one edge over the filling, then gently flip the omelette over. Give it a few more seconds, then slide onto a hot plate while you cook the other one. Serve immediately with hot buttered toast.