Boil the potatoes. Start by cleaning the potatoes under cold water. There is no need to peel them. Next, fill a large pot with water and add the potatoes. Bring to a boil and let simmer for 15-17 minutes depending on the size of your potatoes. The potatoes should be tender but not overcooked. Drain them and let them cool for a few minutes.
Cut them into quarters. Once the potatoes have cooled down, cut them into 1-inch quarters (or into thick slices if using large potatoes). Transfer to a large mixing bowl.
Season. Pour in the melted butter or olive oil and toss to coat. Sprinkle with salt and pepper and toss again.
Prepare your smoker. Remove the lid, rack, and drip tray of your stovetop smoker. Spread 1 tablespoons of wood chips in the bottom of the smoker. Place the rack and drip tray back on top.
Add the potatoes. Arrange the quartered potatoes on top of the rack and close the smoker with its lid.
Smoke. Heat the smoker on the stove over medium heat. It will take 10-15 seconds before the smoke starts to escape from the smoker. At this point, let the potatoes smoke for about 20 minutes, without opening the lid. After that, turn off the heat and let the smoke infuse the potatoes for another 10 minutes.
Let them cool. Let the potatoes cool down for a few more minutes before serving! These potatoes will stay fresh for up to 3 days in the refrigerator.
Notes
Do not overcook the potatoes. Stop boiling the potatoes as soon as they are tender. Otherwise, you will have trouble cutting them into clean quarters. To test doneness, pierce a potato with a knife or a fork. If it pierces easily the center of the potatoes, they are done.
Smoke over medium heat. To prevent the potatoes from burning, smoke them over medium heat.
Let the potatoes rest after smoking. Just like any smoked food, it's best to let the potatoes rest. Doing that allows the flavors to merge and the smokiness to mellow down a little bit.