Heat oven to 400° F. In a medium bowl, toss the potatoes with 2 tablespoons of the oil, ¾ teaspoon salt, and ¼ teaspoon pepper.
Transfer the potatoes to a rimmed baking sheet and roast, stirring once, until golden, about 25 minutes.
Meanwhile, season the steaks with ¾ teaspoon salt and ¼ teaspoon pepper. Heat 1 tablespoon oil in a large skillet over medium heat. Cook to desired doneness, about 4 minutes per side for medium-rare.
Transfer the steaks to a cutting board, cover with foil, and let stand for 5 minutes.
Meanwhile, whisk together the mustard, vinegar, and ¼ teaspoon each salt and pepper. Whisking constantly, slowly add the tarragon and remaining oil.
Toss the spinach with the vinaigrette, slice the steaks, and serve the salad and meat with the potatoes.