Preheat your oven to 400°F (200°C). Lightly grease a standard 12-cup muffin tin with a bit of the melted butter or nonstick spray so the potatoes release easily after baking.
Slice the potatoes into very thin rounds, about 1/8 inch thick or a little less. A mandoline makes quick work of this, but a sharp knife and a steady hand do just fine. Try to keep the slices roughly the same thickness so they cook evenly.
Place the potato slices in a large bowl. Drizzle the melted butter over the potatoes and sprinkle with the salt. Toss with your hands until every slice has a light sheen of butter and a bit of seasoning.
Sprinkle about 2 tablespoons of the shredded cheddar into the bottom of each muffin cup, dividing roughly half of the cheese among the 12 cups. This makes a lovely cheesy base and helps form those lacy, crisp edges.
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