Step 1: Set up your grill for indirect grilling and heat to medium-high.

Step 2: Place each potato on a hasselback cutting guide or between two wooden chopsticks or pencils. Using a sharp knife, cut each potato crosswise into 1/8-inch-wide slices, but don’t cut all the way through; you want the potato to stay intact. Do this right before cooking so the potato won’t oxidize. (The guide or the chopsticks will prevent you from cutting too deeply.) Brush the potatoes with melted butter, forcing it between the slices. Season generously with salt and pepper. If you want to cut the potatoes ahead of time (up to 4 hours), drop them in a bowl of ice water once sliced. There’s an added advantage here: This helps spread the slices apart.

Step 3: Brush or scrape the grill grate clean and oil it well. Place the potatoes on the grill grate over the drip pan away from the heat. Add the wood chunks or chips to the coals. Smoke-roast the potatoes until the slices begin to fan, 30 minutes.

Step 4: Brush the potatoes with melted butter, again forcing it between the slices, and sprinkle with the grated cheese. Continue smoke-roasting until the cheese is melted and lightly browned and the potatoes are tender, 30 to 45 minutes longer. Sprinkle with chives before serving.