Preheat the oven to 450 degrees F.

Place the potatoes on a cutting board. With the tip of a small, sharp knife, score a lid shape on the top of each potato about a third of the way down, cutting in about 1/8 inch all the way around.

Spread a 1/4-inch-thick bed of rock salt on a baking tray. Place the potatoes on the bed of rock salt. Bake the potatoes until tender enough to be pierced easily with a wooden skewer, about 45 minutes. Remove the baking sheet from the oven and leave them for a few minutes to cool slightly. Leave the oven on.

Heat a saute pan over high heat. Add the olive oil and, as soon as it swirls easily, add the butter. When the butter foams, add the ham and mushrooms and saute until they are hot, 2 to 3 minutes. Transfer the ham and mushrooms to a small mixing bowl.

Using a small, sharp knife, and protecting your hand from the heat with a folded kitchen towel, carefully cut off the lids from the still-hot baked potatoes, following the scoring marks. With a sharp-edged tablespoon, scoop the flesh from the potatoes, including the lids, leaving a wall about 1/4-inch thick, and transferring the scooped-out flesh to a large mixing bowl. Return the potato shells and lids to the bed of rock salt. Put the tray back in the hot oven, leaving the oven door open, to keep the potato shells and lids warm.

With a potato masher, mash the potatoes. Season the mixture, to taste, with salt and generously with pepper. Add the creme fraiche or sour cream and the butter and mash to combine them with the potatoes. Stir in the ham, mushrooms, and 2 ounces of the cheese.

Remove the baking sheet from the oven and turn the temperature up to 500 degrees F.

Carefully spoon the mashed potato mixture back into each potato shell. Sprinkle the remaining cheese on top of the filling on each potato. Leave the lids unfilled, placing them around the side of the tray.

Return the tray to the oven and continue baking the potatoes until they are heated through and the cheese has melted, 7 to 10 minutes. If the cheese has not browned, turn on the broiler and broil the potatoes briefly, keeping a close eye on them to prevent burning.

Using tongs, transfer the potatoes to a heated serving platter or individual plates. Place the lids on top of the potatoes at an angle, inserting 1 end partway into the filling to secure them.