Prepare the tapenade by pulsing together the black olives, garlic, anchovies and lemon juice in a food processor.
Meanwhile, lower the eggs gently into a saucepan of boiling water and cook for 12 minutes. Add the olive oil in parts, pulsing in between until a spreading consistency is reached. Transfer to a bowl and reserve to one side.
Drain the eggs after 12 minutes and run under cold water for 1 minute. Peel and slice thinly.
Arrange the sandwiches by spreading the base slice of bread with the tapenade. Top with the tuna evenly. Then place slices of egg and tomato on top.
Finish with a lettuce leaf and the top layer of bread to create the sandwiches. Serve immediately.