In a small bowl, combine the oil, salt and pepper; brush over both sides of eggplant. Place in an ungreased 15x10x1-in. baking pan. Broil 4-6 in. from the heat for 5-7 minutes on each side or until tender and lightly browned.
Meanwhile, place 2-3 in. of water in a large skillet with high sides; add vinegar. Bring to a boil; reduce heat and simmer gently. Break cold eggs, one at a time, into a custard cup or saucer; holding the cup close to the surface of the water, slip each egg into water. Cook, uncovered, until whites are completely set and yolks begin to thicken (but are not hard), about 4 minutes. With a slotted spoon, lift each egg out of the water.
Top each muffin half with an eggplant slice, egg, 3 tablespoons salmon and 1 tablespoon sour cream. Set aside and keep warm.
In a double boiler over simmering water, constantly whisk the egg yolks, water and lemon juice until mixture begins to thicken and coats the back of a metal spoon. Reduce heat to low. Slowly drizzle in melted butter, whisking constantly. Whisk in salt and pepper.
Spoon 3 tablespoons sauce onto eight serving plates. Top with a prepared muffin; sprinkle with chives.