Preheat the oven to 425°F (220°C). Lightly oil a rectangular baking dish.
Put the potatoes in a pot of salted water. Bring to a boil and cook for 8–10 minutes, until just barely fork-tender. Drain thoroughly and let cool enough to handle.
Toss the potatoes with the olive oil and dried rosemary.
Wrap each potato or potato half with a strip of prosciutto. Set the bundles seam-side down in a single layer in the baking dish, leaving a little space between them.
Bake for 20–25 minutes, rotating the dish halfway through, until the potatoes are golden and the prosciutto is crisp at the edges.
Rest for 5 minutes. Transfer to a platter and serve warm.