Scrub the potatoes. If you have a mix of large and small potatoes, cut the large potatoes in half so that everything cooks roughly in the same amount of time.
Add the potatoes to a medium or large pot, and cover with water. Add a generous pinch of salt, and bring to a boil (you must start the potatoes in cold water, or they won’t cook properly). Cook for 20-30 minutes, until fork tender. Drain and cool on a cutting board. If you mix the potato salad when the potatoes are still warm, some of them will “mash.” Fully cooling them will give your potato salad a neater look.
While the potatoes are cooking, you can brown the hot dogs in a skillet with just a small drizzle of oil or grill them until they’re browned. Let the hot dogs cool slightly, then slice them on a diagonal into thick slices.
When the potatoes are cooled, cut them into 1-inch chunks. In a large bowl, whisk the olive oil, dijon mustard, and salt and pepper together. (Because the saltiness of hot dogs can vary by brand, we’ve provided a range for the salt.)
Add the potatoes, shallots, capers, basil, parsley, and chives. Toss everything together, and taste for seasoning. Add more salt and pepper if needed.