Combine sherry vinegar, shallot, and mustard in medium bowl. Gradually whisk in oil. Mix in cheese. Season with salt and pepper.
Cook potatoes in large pot of boiling salted water until not quite tender, approx 10 to 15 minutes (be careful not to overcook or potatoes will crumble when you try to slice them). Drain and cool completely. Slice cooled potatoes into 1/4- to 1/2-inch-thick rounds. Brush rounds on both sides with oil; sprinkle with sea salt and pepper.
Prepare barbecue (medium-high heat). Grill potatoes until golden and cooked through, about 3 minutes per side.
Place arugula on plate and pile warm potatoes on top. Drizzle dressing over potatoes and sprinkle with chopped green onions.