Preheat oven to 375°F (190°C). Lightly brush the cups of a 12-cup muffin tin with oil.
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”.
Place 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.
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.
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.
Serve upside down (with thyme sprig facing up) with a sprinkle of flaky salt.
