Preheat to over 350°F. Prepare muffin pan: cut parchment paper rounds that will line the bottoms of a 12-cup muffin pan.
In a small saucepan, melt butter on medium heat. Brush the muffin pan with butter. Line with parchment paper rounds. Place a small thyme sprig in the center of each round. Add about 1/2 tablespoon of melted butter to each cup.
Chop some thyme leaves, and add them along with the garlic to the remaining butter. Stir on low heat until fragrant.
Use a mandoline to carefully slice potatoes crosswise into thin rounds. Place them in a bowl, pour the butter mixture over top, and toss to coat well.
Divide the potatoes in each muffin cup, layering the rounds in a circular pattern. Sprinkling a little parmesan in between each layer. Press down gently, and drizzle the remaining butter over top.
Cover with foil and bake for 30 minutes, or until you can pierce them easily with a knife.
Using a butter knife, carefully lift each potato stack and placed upside down on a parchment-lined baking sheet, to that the thyme sprig is on top now. Drizzle tops with any remaining butter.
Place back into the oven, uncovered, increase heat to 425°F and bake until the edges are golden and crispy. About 15-20 minutes.