Preheat your oven to 400 degrees Fahrenheit (200 degrees Celsius). Lightly grease a standard 12-cup muffin tin using a little of the melted butter or a quick spray of nonstick cooking spray so the potato stacks will release cleanly and easily once they are done baking.
Slice your potatoes into very thin, even rounds, approximately one-eighth of an inch thick or a touch thinner. A mandoline slicer makes this step go quickly and produces beautifully uniform slices, but a sharp knife and a steady, patient hand will get the job done just as well. Focus on keeping the slices as consistent in thickness as possible so everything cooks evenly throughout.
Transfer all of your potato slices into a large mixing bowl. Drizzle the melted butter evenly over the top and sprinkle the salt across everything. Use your hands to toss the potatoes gently and thoroughly until every single slice has a light, even coating of butter and a touch of seasoning from edge to edge.
Scatter approximately two tablespoons of the shredded cheddar cheese into the bottom of each muffin cup, dividing roughly half of the total cheese evenly among all twelve cups. This creates a wonderful cheesy base layer that helps form those beautiful lacy, deeply golden edges that make these stacks so irresistible.
Now layer the buttered potato slices carefully into each muffin cup, stacking them up to just below the rim of the cup. Press down gently as you go to keep the stacks snug and compact, but do not worry if they look a little imperfect or uneven at this stage — they will settle naturally as they bake and everything will come together beautifully.
Distribute the remaining shredded cheddar evenly across the top of each potato stack, dividing it among all twelve cups. Try to keep the majority of the cheese sitting on top of the potatoes, but a little spilling down the sides is entirely welcome — that is precisely what creates those deeply browned, frilly, crispy edges that everyone fights over on the serving platter.
Cover the muffin tin loosely with aluminum foil, tenting it slightly upward so the foil does not press down directly onto the cheese. Bake covered for 25 minutes, until the potato slices are beginning to turn tender when you pierce them gently with the tip of a sharp knife.
Remove the foil and continue baking uncovered for another 15 to 20 minutes, until the potatoes are completely tender all the way through and the cheese around the edges of each cup has turned deeply golden and beautifully crisp. Keep a watchful eye on them during the final stretch, as ovens vary and you want golden perfection rather than anything burnt.
