When put directly on the grill, potatoes will burn before they are cooked through. Parboiling potatoes and wrapping them in a foil packet with seasonings allows for perfectly grilled potatoes with lots of golden, flavorful bits.
Place potatoes in a large saucepan, and fill with enough cold water to cover them by 2 inches. Bring to a boil over high heat, and add salt. Reduce heat; simmer until potatoes are slightly tender when pierced with a fork (but not cooked through), about 10 minutes. Drain; let stand until cool enough to handle. Cut each potato in half.
Heat grill to high. Overlap two pieces of foil, each about 3 feet long, to form a cross. Place the potatoes in one layer in center of cross. Drizzle the potatoes with 2 tablespoons oil; add the shallots and oregano; season with salt and pepper. Fold foil, enclosing the potatoes, and seal edges by crimping.
Place foil packet on grill, and cook, shaking packet occasionally with tongs, until potatoes are cooked through and golden brown, about 20 minutes, depending on heat of grill. Remove packet from grill, carefully, watching for any steam and hot oil that might escape. Transfer potatoes to a serving bowl. Drizzle with remaining 2 tablespoons oil, and adjust seasoning as desired.