Preheat Oven: Preheat your oven to 375°F (190°C). Grease a 12-cup muffin tin with butter or non-stick spray and set aside.
Prepare Potatoes: Wash and peel the potatoes if desired. Using a mandoline or a sharp knife, slice the potatoes into thin rounds, approximately 1/16 inch (1.6 mm) thick.
Season Potatoes: Place the potato slices in a large mixing bowl. Add the melted butter, grated Parmesan cheese, garlic powder, fresh thyme leaves, salt, and freshly ground black pepper. Toss until all slices are evenly coated with the mixture.
Assemble Stacks: Layer the seasoned potato slices into stacks and place each stack into the prepared muffin cups. Fill each cup to the top, as the potatoes will shrink during baking.
Bake: Place the muffin tin in the preheated oven and bake for 55 minutes to 1 hour, or until the edges and tops of the potato stacks are golden brown and crispy, and the centers are tender when pierced with a knife.
Serve: Once baked, remove the muffin tin from the oven and let the potato stacks rest for about 5 minutes. Carefully remove each stack using a spoon or small spatula. Garnish with additional grated Parmesan cheese and fresh thyme leaves if desired. Serve immediately.