Fill a small pot with water and bring to a boil. Carefully add the eggs to the water with a slotted spoon. Simmer gently for 13 minutes. Transfer the eggs to a bowl of ice water. When cool enough to handle, peel the eggs and dice into 1/4-inch pieces. Set aside.
Put the potato slices in a medium saucepan and cover with cold water by 2 inches. Salt the water heavily. It should taste like seawater. Bring to a boil and then lower to a simmer. Simmer until the potatoes are just cooked through and easily pierced with a knife, about 10 minutes.
While the potatoes are cooking, combine the vinegar and shallots in a large bowl. Let sit for 5 minutes. Whisk in the mustard and then slowly drip in the vegetable oil and then the olive oil while continuing to whisk. Add the chopped capers to the dressing. Season with salt and pepper.
Drain the potatoes and while still warm and add directly to the bowl of dressing. Toss and season with salt and pepper. Let the potatoes come to room temperature before adding the dill, parsley, tarragon and chopped egg. Toss again and serve.