Wrap the grated potatoes in a large clean towel and squeeze out the excess starch. Unwrap the potatoes and place in a large mixing bowl. Season the potatoes with salt and pepper. Stir in the olive oil and mix well. Form the mixture into 12 small cakes about 3 inches in diameter and 1 inch thick. In a large nonstick skillet, over medium heat, heat the vegetable oil. When the oil is hot, pan-fry the cakes, in batches, until golden brown and cooked through, about 4 minutes on each side. Remove the cakes from the pan and drain on a paper-lined plate and keep warm. Pour off the oil and wipe the skillet clean. Melt the butter in the skillet. When the butter has melted, add the onions. Season the onions with pepper. Saute the onions until soft, about 2 minutes. Add the salmon and continue to saute for 1 minute. In a mixing bowl, whisk the eggs, half/half, and green onions together. Season the mixture with salt and pepper. Whisk to blend well. Add the egg mixture to the salmon. Using a plastic spatula, stir the eggs often and cook until the mixture sets slightly, but is still soft, 7 to 8 minutes. Cook 2 to 3 minutes longer if firmer eggs are desired. Add the cheese for about 30 seconds, and remove from heat. Continue stirring the mixture until the cheese melted completely. To serve, top each potato cake with an equal portion of the eggs. Spoon the hollandaise over the top and serve warm.
In a stainless steel bowl set over a pot of simmering water over medium heat, whisk the egg yolks with 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 tough 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.