Put the potatoes in a 6-quart Dutch oven or similar pot and cover with cold water by at least one inch. Add 1-1/2 Tbs. salt, bring to a boil over high heat, and reduce the heat to a gentle simmer. Cook until the potatoes are just tender, 6 to 8 minutes from when the water comes to a boil (stir gently and don't overcook, or the potatoes will fall apart).
Meanwhile, in a bowl or liquid measuring cup, whisk the vinegar and mustard. Whisk in the olive oil. Season to taste with salt and pepper.
Drain the potatoes in a colander and spread them on a rimmed baking sheet. While they're still hot, drizzle them evenly with 3 Tbs. of the dressing. Let cool completely.
Transfer the cooled potatoes to a serving bowl. Sprinkle with the olives, scallions, parsley, and mint. Pour about 1/2 cup of the remaining vinaigrette over the salad. With a large spoon or rubber spatula, gently toss. Take care to break as few of the potato slices as possible. Season to taste with salt and pepper. Let sit at room temperature for at least an hour so the flavors can blend.
Just before serving, season to taste again with salt and pepper and add more dressing if the salad seems dry—you may not need all of the dressing.
You can make this salad up to 12 hours ahead; just cover and refrigerate, and return to room temperature before serving.