Place the potatoes* in a large pot and cover with cold water. Generously season with salt, place over high heat, and bring to a boil. Once the water comes to a boil, reduce the heat to medium and let the potatoes simmer for 15-20 minutes, until fork-tender
Drain the potatoes and set aside to cool slightly.
Meanwhile, whisk together the Dijon mustard, lemon juice, apple cider vinegar, and olive oil in a small bowl until smooth and creamy. Season the dressing with salt and pepper to taste.
Once the potatoes are cool enough to handle, peel them**, and depending on their size, either halve or quarter them.
Add the potatoes to a large mixing bowl and toss in the scallions, chives, and capers while the potatoes are still hot. Pour in the tangy dressing and toss gently to evenly coat the potatoes.
Layer the arugula on a serving platter and top with the potato salad. Nestle in the sliced radishes and serve immediately. Garnish with very thinly sliced lemon slices and additional chives if desired.