Place potatoes in a medium saucepan and cover with cold salted water by 2 inches. Cover and bring to a boil, reduce heat and simmer until tender, 8 to 10 minutes. Drain and toss with ⅓ cup vinaigrette in a bowl. Let sit, stirring occasionally, until the dressing is absorbed.
Meanwhile, heat oil in a heavy bottomed skillet over medium-high heat. Season steak with 1 teaspoon salt and ½ teaspoon pepper. Cook, partially covered, until the internal temperature registers 120-125°, about 3 minutes per side, or to desired degree of doneness. Remove to a plate and let rest 5 minutes before slicing.
Add arugula, watercress, scallions, dill, and remaining vinaigrette to potatoes; gently toss to combine. Season with salt and pepper.
Serve with steak and sprinkle with goat cheese.
To make the champagne vinaigrette, whisk together ¼ cup olive oil, 2 tablespoons Champagne vinegar, 1 ½ teaspoons whole-grain Dijon mustard, 1 minced small garlic clove, and ¾ teaspoon sugar. Season with salt and freshly ground black pepper.