Step 1Place the potatoes in a medium saucepan with the bay leaves, garlic and 1 teaspoon salt. Add water to cover and bring to a boil. Reduce the heat and simmer until the potatoes are easily pierced with a knife, about 15 minutes (time will vary depending on type and age of potatoes). Drain well. Cool slightly.
Step 2While the potatoes cook, place the eggs in a small saucepan with water to cover. Add one-half teaspoon salt and the cider vinegar. Bring to a boil, reduce the heat and simmer for 10 minutes. Drain and place under cold running water until cool.
Step 3Slice the potatoes crosswise into half-inch pieces and place in a bowl. Add the chopped anchovies. Cut the piquillos in half lengthwise, then crosswise into thin strips. Add them to the potatoes. Cut the shallot in half lengthwise, then crosswise into very fine slices and add.
Step 4Cut the eggs in half lengthwise and scoop out the yolks. Press the yolks through a small strainer into a small bowl. Set aside. Cut the whites in half lengthwise and then crosswise into half-inch thick strips and add to the potatoes. Toss the potato mixture with a fork until the ingredients are well mixed.
Step 5To the yolks, add the mayonnaise, vinegar and olive oil and mix gently until smooth and well combined. Pour the dressing over the potato mixture. Mix together with a rubber spatula until the ingredients are well distributed and lightly dressed. Season with salt and pepper to taste. Add more chopped anchovies if desired.
Step 6To serve, lay the arugula in a salad bowl or on serving plates and mound the potato salad over.