Cut a small piece off the bottoms of the potatoes so they do not roll around.
Place two wooden spoons alongside the potatoes and slice them vertically every 1/8" down to the spoons.
Microwave potatoes for 5 minutes. Flip and microwave another 5 minutes. Let cool until you can handle them.
Grabbing the ends of the potatoes, brush them with oil, getting some oil between the slices.
Sprinkle garlic into the slices.
Sprinkle tops of potatoes with the Parmesan, thyme, and salt and pepper. Cover loosely with foil.
Place over indirect heat in a preheated grill until tender, about 20 minutes.
Serve sprinkled with the remaining mozzarella cheese.