Preheat the oven to 400°F. Wrap the potatoes in aluminum foil and bake for about 1 hour, until very tender. Remove from the oven and let cool slightly. When cool enough to handle, cut each potato in half lengthwise and scoop out the flesh, leaving a thin layer of flesh on the skin of each potato. Reserve the potato flesh for another use.
While the potatoes are cooking, arrange the prosciutto slices on a small rimmed baking sheet. Bake for 10 to 12 minutes, until crispy. When cool enough to handle, crumble the prosciutto and set aside.
Brush the olive oil over both sides of the potato skins and season liberally with salt and pepper. Place skin side up on a sheet pan and cook for 5 minutes. Flip the potatoes and cook for another 3 to 4 minutes, until the skins are crispy and lightly browned.
Sprinkle the cheese over the potato skins and return to the oven for about 2 minutes, until the cheese is melted. Remove from the oven and top the potato skins with the crème fraiche, crumbled prosciutto, chives, and more pepper.