Prepare a grill for indirect heat. If using a charcoal grill, build the hot coals on one side only. If using a gas grill, heat one side only to medium-high heat.
Place a heavy-bottomed Dutch oven over direct heat. Add a drizzle of olive oil along with the potatoes. Season with salt and pepper and add the garlic. Cook, stirring occasionally, until golden brown on all sides, 5 to 7 minutes. Add enough water to cover the potatoes halfway, close the grill and steam until the potatoes are cooked through and the water has evaporated, 10 to 12 minutes.
Lightly smash the potatoes with a potato masher and drizzle with more olive oil. Cook until the potatoes are crispy on the bottom, about 5 minutes. Serve sprinkled with flakey salt and dill and a squeeze of lemon.
(Alternatively, you can cook the potatoes over medium-high heat on a stovetop.)