Preheat oven to 375°F. Lightly brush the cups of a 12-cup muffin tin with oil. Pile stacks up high for 12 slightly heaped stacks, keeping in mind the potatoes sink down as they cook.
Melt butter and stir in garlic, parsley, chopped thyme, salt and black pepper. Set aside to cool
Slice potatoes with skin on, very thinly about 1/16”. I highly recommend using a mandolin to get super thin slices of potatoes in relatively little time. Evenly-thick slices are important for even cooking. If you don’t have a mandolin, slice carefully with a sharp knife. Place sliced potatoes in a large mixing bowl and toss with cooled garlic herb butter.
Assemble potato stacks. In each oiled muffin cup, place a couple rosemary leaves followed by slices of potatoes starting with smaller pieces and ending with the larger pieces. About 1/3 and 2/3 of the way up the stack, sprinkle a bit of grated parmesan. For the prettiest stacks, stagger potato slices a little rather than matching them up in a neat column. Divide evenly among the 12 cups.
Cover muffin tin tightly with foil. Bake in preheated 375°F (190°C) oven for 25 minutes or whenever potatoes are tender (use a wooden skewer to poke one stack to check – the skewer should easily slide all the way through and back out). Increase oven heat to 425°F (220°C). Remove foil and return to bake uncovered for 13-15 minutes more, or whenever edges are browned and crispy, and tops are golden brown. Keep a close eye on them during the final few minutes of browning.
Dislodge the stacks from the muffin tin with a fork. Serve upside down with a sprinkle of Maldon Sea Salt Flakes.