Preheat the oven to 375°F/190°C.
Wash the potatoes and place them in a 9x13-in. baking pan. Drizzle the potatoes with 1 tsp. of the oil, season with salt and pepper, cover with foil and bake until tender, about 50 - 60 min.
Preheat a ridged grill pan or an outdoor grill.
When the potatoes are cool enough to handle, cut them in half. Place the potatoes in a bowl and drizzle them with 2 Tbsp. of the olive oil. Toss the potatoes to coat them.
Grill them, turning occasionally until golden brown, about 5 - 7 minutes.
Remove the potatoes from the grill and place them in a large serving bowl. Add the remaining olive oil, green onions, garlic, parsley, oregano and olives. Season to taste with salt and pepper.
Using a cheese shaver, shave long, thin slices of the Parmigiano on top of the potatoes. Toss gently and serve immediately.