Bring two medium saucepans of salted water to a boil.
Gently lower the eggs into the boiling water of one saucepan and simmer gently, uncovered, for 10 minutes. Remove the eggs to a bowl of ice water. Once cool, peel and quarter the eggs.
Add the haricots verts to the second saucepan and blanch until just tender, about 3 minutes. Cool under running water, drain, and pat dry.
In a medium bowl, whisk together the vinegar and mustard. Season with salt and pepper. Whisk in the oil in a slow, steady stream to make a smooth, emulsified vinaigrette.
Slice the warm potatoes and then toss in a medium bowl with a few tablespoons of the dressing. Season with salt and pepper.
Line a serving platter with the lettuce. Mound the potatoes on the lettuce. Toss the haricots verts with a few tablespoons of dressing, season with salt and pepper, and mound next to the potatoes. Repeat with the tomatoes and tuna, dressing separately and placing on the lettuce. Add the egg quarters, but don't toss with the dressing. Arrange the anchovies between the mounds of ingredients and scatter the olives. Drizzle a little more of the dressing over everything and serve family style, with any extra dressing on the side.
To cook the new red potatoes, put the potatoes in a saucepan with water to cover by 1 inch. Bring to a simmer and cook until tender throughout, about 15 minutes. Drain and let cool slightly.