Preheat your oven to 375°F (190°C).
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 and a fork to hold the potato in place safely.
Place the sliced potatoes in a large mixing bowl and toss with cooled garlic herb butter.
In each oiled muffin cup, place a small sprig of thyme, followed by slices of potatoes starting with smaller pieces and ending with the larger pieces. About ⅓ and ⅔ 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. It’s okay if the potato stacks are a little taller than the rim of the muffin cups because they will sink down as they cook.
Cover the 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. See recipe notes for make ahead tips.
I use the same wooden skewer to dislodge the stacks from the muffin tin by gently pushing the tip down one side and flipping it enough to grab it with tongs. Serve upside down (with thyme sprig facing up) with a sprinkle of flaky salt. Enjoy!