Toss potatoes with 1 Tbsp oil and salt and pepper, roast on a rimmed parchment-lined baking sheet at 400°F for 20 minutes or until golden and tender, tossing occasionally.
To boil eggs, place eggs in a single layer in a saucepan adding enough (cold) water to cover by 1 inch. Cover and bring to just a boil. Remove from heat and let sit in the hot water (covered) for 12 minutes. Gently crack eggs in a few places then place immediately in a bowl of cold water for at least 5 minutes.
For the dressing, stir together the vinegar and mustard and season with salt and pepper. While whisking constantly, drizzle in the oil until a thick emulsion is formed.
Once potatoes are done, pour some of the dressing over the warm potatoes while still on the baking sheet and toss to combine.
Add the arugula to the pan, and toss to combine (the heat will lightly wilt the leaves and warm the salad nicely).
Divide arugula, potatoes and eggs into individual serving bowls, drizzling on more dressing if desired.