Prep: Preheat oven to 400°F (200°C). Grease a 12-cup muffin tin with butter or nonstick spray.
Slice: Slice potatoes into very thin rounds (approx. 1/8 inch thick). Using a mandoline is recommended for uniformity.
Toss: In a large bowl, toss the potato slices with melted butter and salt until every slice is evenly coated.
Base Layer: Sprinkle about 2 tsp of cheese into the bottom of each muffin cup (using about half the cheese total) to create a crispy "crust."
Stack: Layer the buttered slices into the cups, stacking them just below the rim. Press down gently to pack them in.
Top: Divide the remaining cheese evenly over the tops of the stacks.
Initial Bake: Cover loosely with foil (tent it so it doesn't touch the cheese). Bake for 25 minutes.
Final Bake: Remove foil and bake for another 15–20 minutes until the centers are tender and the edges are deeply golden and crisp.
Rest & Release: Let rest in the pan for 5 minutes. Run a small knife around the edges to loosen and lift out with a spoon.