Step 1Combine the mustard and lemon juice in a small bowl and whisk to blend. Whisk in the olive oil until smooth, then the mayonnaise. Stir in the capers and chives and set aside.
Step 2Combine the potatoes and salt in a medium pot and add cold water to cover. Bring it to a boil and cook until the potatoes are easily pierced with a knife, about 10 to 15 minutes. Remove the potatoes with a slotted spoon to a colander to drain.
Step 3Add the green beans to the water and cook until just tender, about 2 to 4 minutes. Drain well.
Step 4While the potatoes are still warm, cut them lengthwise into quarters. In a bowl, combine the potatoes with the olives. Add half the dressing and toss to coat. Season with salt and pepper to taste. Arrange in the center of a platter.
Step 5Toss the green beans with more of the dressing in the same bowl. Add the peppers, season with salt and pepper and toss to mix. Arrange along one side of the potatoes.
Step 6Arrange the eggs and tomatoes on either side of the other vegetables. Drape the anchovies decoratively over the potatoes and beans and serve with the remaining dressing on the side.