In a large mixing bowl, whisk the eggs and milk together and season with salt and pepper. In a 6-inch nonstick saute pan over medium heat, heat 1 tablespoon of the butter. Add 1 tablespoon of the red onions.
Pour in one-fourth of the egg-cream mixture and cook, shaking the pan and stirring rapidly with a fork, holding the tines parallel to the bottom of the pan. The omelet should remain runny in the center but firm on the bottom. Lift the handle of the skillet and gently knock the pan on the surface of the stove so the omelette slides to the bottom curve of the pan.
Add 1/4 cup of the salmon and 1/4 cup of the cheese in the center of the omelet, then with a fork, quickly fold the omelette from the top down. Let it stand for a few moments, then slide it onto a serving plate.
Repeat the process until all the egg mixture and butter are used. To serve, spoon the sauce over each omelet and garnish with parsley.
In a stainless steel bowl set over a pot of simmering water over medium heat, whisk the egg yolks with the lemon juice, water, mustard, and parsley, together. Season with salt and cayenne. Whisk the mixture until pale yellow and slightly thick. Be careful not to let the bowl touch the water. Remove the bowl from the pot and whisking vigorously, add the butter, 1 teaspoon at a time, until all is incorporated. Fold in the caviar and serve warm.