In a medium pot, add the potatoes and enough cold water to cover by an inch. Bring to a boil and cook until just tender, 13 minutes. About 5 minutes before the potatoes are done, add the string beans and finish cooking with the potatoes. Drain in a colander, then rinse with cold water.
Meanwhile, in a medium bowl, whisk together the mustard, lemon juice, vinegar, 1/2 teaspoon salt and 1/4 teaspoon pepper. Slowly whisk in the olive oil to make a vinaigrette; stir in the tarragon and parsley.
In a medium bowl, combine the tuna with the vinaigrette. Coarsely chop the green beans, olives and half of the capers, then stir into the tuna. Coarsely chop the eggs and fold them into the tuna mixture; season with salt and pepper. Cut the ends off the potatoes and discard. Cut each potato into 4 rounds and arrange on a platter. Season each potato round with salt and pepper, then top with the tuna mixture and the remaining capers.