Bring a small saucepan full of water to a boil. Reduce the water to a simmer and gently place the eggs into the water. Simmer for 10 minutes. Remove and place the eggs into a bowl of ice water to cool. After about 5 minutes in the cool water, remove the eggs and peel the shells. Cut each egg in half lengthwise and remove the cooked yolks. Add the yolk to a small bowl and reserve the whites.
Finely chop the purple and light green part of the onion and add to the yolks. Slice the top, bright green parts and reserve for the garnish. Finely chop the salmon, saving about ¼ of a piece for garnish, if desired. Add the salmon, mayo, olive oil, vinegar, lemon juice and a few grinds of fresh cracked pepper to the bowl. With a fork, mix until thoroughly combined. Season to taste. Spoon the mixture into the egg whites and garnish with the extra, small salmon pieces and chopped green onions. You can also put the yolk mixture into a ziploc bag and cut off the corner to create a makeshift pastry bag. Then you can pipe the mixture into the egg whites for a more refined presentation.