Bring 1 quart of water to a boil. Place eggs in boiling water and let simmer for 9 minutes. Remove and place into ice water to cool down. Peel and cut each egg into quarters.
Place the bread slices on a baking sheet and drizzle with the extra virgin olive oil. Place the baking sheet into the preheated oven with the broiler setting and cook until toasted and golden brown.
Remove and spread the aioli or mayonnaise evenly between the toasts. Flake the tuna on the crostini, then place the cherry tomatoes and egg wedges over the tuna and sprinkle lightly with sea salt. Drizzle a little olive tapenade over the toast and garnish with basil.
Finely chop olives, anchovies and capers. Place in a bowl and whisk in the olive oil.
Put the lemon juice, egg yolks, garlic and salt in a blender; blend until completely combined. Combine the oils in a cup; with the blender on (let it run for a few seconds before you carefully remove the top), pour the oil in a slow, steady stream and continue to blend. It will thicken and is done when it looks like pudding.