Step 1Cut any large potatoes in half to speed up the cooking. Place the potatoes in a large saucepan, cover them with water by about 1/2 inch and add salt to taste. Bring to a boil. Cover and simmer over medium-low heat until a knife can pierce the center of the largest potato easily and the potato falls from the knife when lifted, about 25 minutes.
Step 2Meanwhile prepare the dressing. Whisk 2 tablespoons of lemon juice with a pinch of salt, pepper and cayenne pepper to taste in a small bowl. Add 3 tablespoons of olive oil and whisk again.
Step 3Drain the potatoes in a colander, rinse them briefly and leave just until cool enough to handle. Peel them if you like, and cut them in 1-inch dice.
Step 4Place them in a large bowl and add the onion, 3 tablespoons of dressing and a sprinkling of salt and pepper. With a rubber spatula, fold together gently but thoroughly. Let the potatoes cool to room temperature.
Step 5Dice the tomatoes (to add them to the salad), or cut them in thick wedges to use as garnish. Whisk the remaining dressing, pour it over the potatoes and fold it in. Add the parsley, capers, half the olives and the diced tomatoes to the potatoes and fold together lightly. Taste, adjust seasoning, and add more lemon juice and olive oil if needed. Garnish with the remaining olives and with the tomatoes, if you've cut them in wedges. Serve at room temperature.